{"id":785,"date":"2025-06-20T17:12:33","date_gmt":"2025-06-20T17:12:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=785"},"modified":"2025-08-13T15:51:41","modified_gmt":"2025-08-13T15:51:41","slug":"numerical-and-improper-integration-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/numerical-and-improper-integration-get-stronger\/","title":{"raw":"Numerical and Improper Integration: Get Stronger","rendered":"Numerical and Improper Integration: Get Stronger"},"content":{"raw":"

Numerical Integration Methods<\/span><\/h2>\r\n

In the following exercises (1-4), approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson's rule as indicated. (Round answers to three decimal places.)<\/strong><\/p>\r\n\r\n

    \r\n \t
  1. [latex]{\\displaystyle\\int }_{1}^{2}\\dfrac{dx}{x}[\/latex]; trapezoidal rule; [latex]n=5[\/latex]<\/li>\r\n \t
  2. [latex]{\\displaystyle\\int }_{0}^{3}\\sqrt{4+{x}^{3}}dx[\/latex]; Simpson's rule; [latex]n=3[\/latex]<\/li>\r\n \t
  3. [latex]{\\displaystyle\\int }_{0}^{1}{\\sin}^{2}\\left(\\pi x\\right)dx[\/latex]; midpoint rule; [latex]n=3[\/latex]<\/li>\r\n \t
  4. Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{2}^{4}{x}^{2}dx[\/latex].<\/li>\r\n<\/ol>\r\n

    In the following exercises (5-9), approximate the integral to three decimal places using the indicated rule.<\/strong><\/p>\r\n\r\n

      \r\n \t
    1. [latex]{\\displaystyle\\int }_{0}^{1}{\\sin}^{2}\\left(\\pi x\\right)dx[\/latex]; trapezoidal rule; [latex]n=6[\/latex]<\/li>\r\n \t
    2. [latex]{\\displaystyle\\int }_{0}^{3}\\dfrac{1}{1+{x}^{3}}dx[\/latex]; Simpson's rule; [latex]n=3[\/latex]<\/li>\r\n \t
    3. [latex]{\\displaystyle\\int }_{0}^{0.8}{e}^{\\text{-}{x}^{2}}dx[\/latex]; Simpson's rule; [latex]n=4[\/latex]<\/li>\r\n \t
    4. [latex]{\\displaystyle\\int }_{0}^{0.4}\\sin\\left({x}^{2}\\right)dx[\/latex]; Simpson's rule; [latex]n=4[\/latex]<\/li>\r\n \t
    5. [latex]{\\displaystyle\\int }_{0.1}^{0.5}\\dfrac{\\cos{x}}{x}dx[\/latex]; Simpson's rule; [latex]n=4[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises (10-20), solve each problem.<\/strong>\r\n
        \r\n \t
      1. Approximate [latex]{\\displaystyle\\int }_{2}^{4}\\dfrac{1}{\\text{ln}x}dx[\/latex] using the midpoint rule with four subdivisions to four decimal places.<\/li>\r\n \t
      2. Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{0}^{0.8}{x}^{3}dx[\/latex] to four decimal places.<\/li>\r\n \t
      3. Using Simpson's rule with four subdivisions, find [latex]{\\displaystyle\\int }_{0}^{\\dfrac{\\pi}{2}}\\cos\\left(x\\right)dx[\/latex].<\/li>\r\n \t
      4. Given [latex]{\\displaystyle\\int }_{0}^{1}x{e}^{\\text{-}x}dx=1-\\dfrac{2}{e}[\/latex], use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.<\/li>\r\n \t
      5. Use Simpson's rule with four subdivisions to approximate the area under the probability density function [latex]y=\\dfrac{1}{\\sqrt{2\\pi }}{e}^{\\dfrac{\\text{-}{x}^{2}}{2}}[\/latex] from [latex]x=0[\/latex] to [latex]x=0.4[\/latex].<\/li>\r\n \t
      6. The length of one arch of the curve [latex]y=3\\sin\\left(2x\\right)[\/latex] is given by [latex]L={\\displaystyle\\int }_{0}^{\\dfrac{\\pi}{2}}\\sqrt{1+36{\\cos}^{2}\\left(2x\\right)}dx[\/latex]. Estimate L using the trapezoidal rule with [latex]n=6[\/latex].<\/li>\r\n \t
      7. Estimate the area of the surface generated by revolving the curve [latex]y=\\cos\\left(2x\\right),0\\le x\\le \\dfrac{\\pi }{4}[\/latex] about the x-axis. Use the trapezoidal rule with six subdivisions.<\/li>\r\n \t
      8. The growth rate of a certain tree (in feet) is given by [latex]y=\\dfrac{2}{t+1}+{e}^{\\dfrac{\\text{-}{t}^{2}}{2}}[\/latex], where t is time in years. Estimate the growth of the tree through the end of the second year by using Simpson's rule, using two subintervals. (Round the answer to the nearest hundredth.)<\/li>\r\n \t
      9. Given [latex]{\\displaystyle\\int }_{1}^{5}\\left(3{x}^{2}-2x\\right)dx=100[\/latex], approximate the value of this integral using the trapezoidal rule with 16 subdivisions and determine the absolute error.<\/li>\r\n \t
      10. The table represents the coordinates [latex]\\left(x,\\text{ }y\\right)[\/latex] that give the boundary of a lot. The units of measurement are meters. Use the trapezoidal rule to estimate the number of square meters of land that is in this lot.<\/li>\r\n<\/ol>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        [latex]x[\/latex]<\/th>\r\n[latex]y[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
        [latex]0[\/latex]<\/td>\r\n[latex]125[\/latex]<\/td>\r\n[latex]600[\/latex]<\/td>\r\n[latex]95[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]100[\/latex]<\/td>\r\n[latex]125[\/latex]<\/td>\r\n[latex]700[\/latex]<\/td>\r\n[latex]88[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]200[\/latex]<\/td>\r\n[latex]120[\/latex]<\/td>\r\n[latex]800[\/latex]<\/td>\r\n[latex]75[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]300[\/latex]<\/td>\r\n[latex]112[\/latex]<\/td>\r\n[latex]900[\/latex]<\/td>\r\n[latex]35[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]400[\/latex]<\/td>\r\n[latex]90[\/latex]<\/td>\r\n[latex]1000[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]500[\/latex]<\/td>\r\n[latex]90[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
          \r\n \t
        1. The \"Simpson\" sum is based on the area under a ____.<\/li>\r\n<\/ol>\r\n

          Error Analysis in Numerical Integration<\/span><\/h2>\r\n

          For the following exercises (1-2), find an upper bound for the error in estimating the given integral using the specified numerical integration method.<\/strong><\/p>\r\n\r\n

            \r\n \t
          1. Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{4}^{5}\\dfrac{1}{{\\left(x - 1\\right)}^{2}}dx[\/latex] using the trapezoidal rule with seven subdivisions.<\/li>\r\n \t
          2. Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{2}^{5}\\dfrac{1}{x - 1}dx[\/latex] using Simpson's rule with [latex]n=10[\/latex] steps.<\/li>\r\n<\/ol>\r\n

            For the following exercises (3-4), estimate the minimum number of subintervals needed to approximate the given integral with the specified error tolerance using the trapezoidal rule.<\/strong><\/p>\r\n\r\n

              \r\n \t
            1. Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{1}^{4}\\left(5{x}^{2}+8\\right)dx[\/latex] with an error magnitude of less than [latex]0.0001[\/latex] using the trapezoidal rule.<\/li>\r\n \t
            2. Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{2}^{3}\\left(2{x}^{3}+4x\\right)dx[\/latex] with an error of magnitude less than [latex]0.0001[\/latex] using the trapezoidal rule.<\/li>\r\n<\/ol>\r\n

              Improper Integrals<\/span><\/h2>\r\n

              For the following exercises (1-4), evaluate the following integrals. If the integral is not convergent, answer \"divergent.\"<\/strong><\/p>\r\n\r\n

                \r\n \t
              1. [latex]{\\displaystyle\\int }_{2}^{4}\\dfrac{dx}{{\\left(x - 3\\right)}^{2}}[\/latex]<\/li>\r\n \t
              2. [latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{1}{\\sqrt{4-{x}^{2}}}dx[\/latex]<\/li>\r\n \t
              3. [latex]{\\displaystyle\\int }_{1}^{\\infty }x{e}^{\\text{-}x}dx[\/latex]<\/li>\r\n \t
              4. Without integrating, determine whether the integral [latex]{\\displaystyle\\int }_{1}^{\\infty }\\dfrac{1}{\\sqrt{{x}^{3}+1}}dx[\/latex] converges or diverges by comparing the function [latex]f\\left(x\\right)=\\dfrac{1}{\\sqrt{{x}^{3}+1}}[\/latex] with [latex]g\\left(x\\right)=\\dfrac{1}{\\sqrt{{x}^{3}}}[\/latex].<\/li>\r\n<\/ol>\r\n

                For the following exercises (5-13), determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.<\/strong><\/p>\r\n\r\n

                  \r\n \t
                1. [latex]{\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}x}\\cos{x}dx[\/latex]<\/li>\r\n \t
                2. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{\\text{ln}x}{\\sqrt{x}}dx[\/latex]<\/li>\r\n \t
                3. [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\dfrac{1}{{x}^{2}+1}dx[\/latex]<\/li>\r\n \t
                4. [latex]{\\displaystyle\\int }_{-2}^{2}\\dfrac{dx}{{\\left(1+x\\right)}^{2}}[\/latex]<\/li>\r\n \t
                5. [latex]{\\displaystyle\\int }_{0}^{\\infty }\\sin{x}dx[\/latex]<\/li>\r\n \t
                6. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dx}{\\sqrt[3]{x}}[\/latex]<\/li>\r\n \t
                7. [latex]{\\displaystyle\\int }_{-1}^{2}\\dfrac{dx}{{x}^{3}}[\/latex]<\/li>\r\n \t
                8. [latex]{\\displaystyle\\int }_{0}^{3}\\dfrac{1}{x - 1}dx[\/latex]<\/li>\r\n \t
                9. [latex]{\\displaystyle\\int }_{3}^{5}\\dfrac{5}{{\\left(x - 4\\right)}^{2}}dx[\/latex]<\/li>\r\n<\/ol>\r\n

                  In the following exercise, determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.<\/strong><\/p>\r\n\r\n

                    \r\n \t
                  1. [latex]{\\displaystyle\\int }{1}^{\\infty }\\dfrac{dx}{\\sqrt{x}+1}[\/latex]; compare with [latex]{\\displaystyle\\int }{1}^{\\infty }\\dfrac{dx}{2\\sqrt{x}}[\/latex].<\/li>\r\n<\/ol>\r\n

                    For the following exercises (15-19), evaluate the integrals. If the integral diverges, answer \"diverges.\"<\/strong><\/p>\r\n\r\n

                      \r\n \t
                    1. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dx}{{x}^{\\pi }}[\/latex]<\/li>\r\n \t
                    2. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dx}{1-x}[\/latex]<\/li>\r\n \t
                    3. [latex]{\\displaystyle\\int }_{-1}^{1}\\dfrac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\r\n \t
                    4. [latex]{\\displaystyle\\int }_{0}^{e}\\text{ln}\\left(x\\right)dx[\/latex]<\/li>\r\n \t
                    5. [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\dfrac{x}{{\\left({x}^{2}+1\\right)}^{2}}dx[\/latex]<\/li>\r\n<\/ol>\r\n

                      For the following exercises (20-27), evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.<\/strong><\/p>\r\n\r\n

                        \r\n \t
                      1. [latex]{\\displaystyle\\int }_{0}^{9}\\dfrac{dx}{\\sqrt{9-x}}[\/latex]<\/li>\r\n \t
                      2. [latex]{\\displaystyle\\int }_{0}^{3}\\dfrac{dx}{\\sqrt{9-{x}^{2}}}[\/latex]<\/li>\r\n \t
                      3. [latex]{\\displaystyle\\int }_{0}^{4}x\\text{ln}\\left(4x\\right)dx[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises (23-27), solve each problem.<\/strong>\r\n
                          \r\n \t
                        1. Evaluate [latex]{\\displaystyle\\int }_{.5}^{1}\\dfrac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]. (Be careful!) (Express your answer using three decimal places.)<\/li>\r\n \t
                        2. Evaluate [latex]{\\displaystyle\\int }_{2}^{\\infty }\\dfrac{dx}{{\\left({x}^{2}-1\\right)}^{\\dfrac{3}{2}}}[\/latex].<\/li>\r\n \t
                        3. Find the area of the region bounded by the curve [latex]y=\\dfrac{7}{{x}^{2}}[\/latex], the x-axis, and on the left by [latex]x=1[\/latex].<\/li>\r\n \t
                        4. Find the area under [latex]y=\\dfrac{5}{1+{x}^{2}}[\/latex] in the first quadrant.<\/li>\r\n \t
                        5. Find the volume of the solid generated by revolving about the y-axis the region under the curve [latex]y=6{e}^{-2x}[\/latex] in the first quadrant.<\/li>\r\n<\/ol>\r\n

                          The Laplace transform of a continuous function over the interval [latex]\\left[0,\\infty \\right)[\/latex] is defined by [latex]F\\left(s\\right)={\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}sx}f\\left(x\\right)dx[\/latex]. This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of [latex] F[\/latex] is the set of all real numbers [latex]s[\/latex] such that the improper integral converges. Find the Laplace transform [latex]F [\/latex] of each of the following functions (28-30) and give the domain of [latex]F[\/latex].<\/strong><\/p>\r\n\r\n

                            \r\n \t
                          1. [latex]f\\left(x\\right)=1[\/latex]<\/li>\r\n \t
                          2. [latex]f\\left(x\\right)=\\cos\\left(2x\\right)[\/latex]<\/li>\r\n<\/ol>","rendered":"

                            Numerical Integration Methods<\/span><\/h2>\n

                            In the following exercises (1-4), approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)<\/strong><\/p>\n

                              \n
                            1. [latex]{\\displaystyle\\int }_{1}^{2}\\dfrac{dx}{x}[\/latex]; trapezoidal rule; [latex]n=5[\/latex]<\/li>\n
                            2. [latex]{\\displaystyle\\int }_{0}^{3}\\sqrt{4+{x}^{3}}dx[\/latex]; Simpson’s rule; [latex]n=3[\/latex]<\/li>\n
                            3. [latex]{\\displaystyle\\int }_{0}^{1}{\\sin}^{2}\\left(\\pi x\\right)dx[\/latex]; midpoint rule; [latex]n=3[\/latex]<\/li>\n
                            4. Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{2}^{4}{x}^{2}dx[\/latex].<\/li>\n<\/ol>\n

                              In the following exercises (5-9), approximate the integral to three decimal places using the indicated rule.<\/strong><\/p>\n

                                \n
                              1. [latex]{\\displaystyle\\int }_{0}^{1}{\\sin}^{2}\\left(\\pi x\\right)dx[\/latex]; trapezoidal rule; [latex]n=6[\/latex]<\/li>\n
                              2. [latex]{\\displaystyle\\int }_{0}^{3}\\dfrac{1}{1+{x}^{3}}dx[\/latex]; Simpson’s rule; [latex]n=3[\/latex]<\/li>\n
                              3. [latex]{\\displaystyle\\int }_{0}^{0.8}{e}^{\\text{-}{x}^{2}}dx[\/latex]; Simpson’s rule; [latex]n=4[\/latex]<\/li>\n
                              4. [latex]{\\displaystyle\\int }_{0}^{0.4}\\sin\\left({x}^{2}\\right)dx[\/latex]; Simpson’s rule; [latex]n=4[\/latex]<\/li>\n
                              5. [latex]{\\displaystyle\\int }_{0.1}^{0.5}\\dfrac{\\cos{x}}{x}dx[\/latex]; Simpson’s rule; [latex]n=4[\/latex]<\/li>\n<\/ol>\n

                                For the following exercises (10-20), solve each problem.<\/strong><\/p>\n

                                  \n
                                1. Approximate [latex]{\\displaystyle\\int }_{2}^{4}\\dfrac{1}{\\text{ln}x}dx[\/latex] using the midpoint rule with four subdivisions to four decimal places.<\/li>\n
                                2. Use the trapezoidal rule with four subdivisions to estimate [latex]{\\displaystyle\\int }_{0}^{0.8}{x}^{3}dx[\/latex] to four decimal places.<\/li>\n
                                3. Using Simpson’s rule with four subdivisions, find [latex]{\\displaystyle\\int }_{0}^{\\dfrac{\\pi}{2}}\\cos\\left(x\\right)dx[\/latex].<\/li>\n
                                4. Given [latex]{\\displaystyle\\int }_{0}^{1}x{e}^{\\text{-}x}dx=1-\\dfrac{2}{e}[\/latex], use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.<\/li>\n
                                5. Use Simpson’s rule with four subdivisions to approximate the area under the probability density function [latex]y=\\dfrac{1}{\\sqrt{2\\pi }}{e}^{\\dfrac{\\text{-}{x}^{2}}{2}}[\/latex] from [latex]x=0[\/latex] to [latex]x=0.4[\/latex].<\/li>\n
                                6. The length of one arch of the curve [latex]y=3\\sin\\left(2x\\right)[\/latex] is given by [latex]L={\\displaystyle\\int }_{0}^{\\dfrac{\\pi}{2}}\\sqrt{1+36{\\cos}^{2}\\left(2x\\right)}dx[\/latex]. Estimate L using the trapezoidal rule with [latex]n=6[\/latex].<\/li>\n
                                7. Estimate the area of the surface generated by revolving the curve [latex]y=\\cos\\left(2x\\right),0\\le x\\le \\dfrac{\\pi }{4}[\/latex] about the x-axis. Use the trapezoidal rule with six subdivisions.<\/li>\n
                                8. The growth rate of a certain tree (in feet) is given by [latex]y=\\dfrac{2}{t+1}+{e}^{\\dfrac{\\text{-}{t}^{2}}{2}}[\/latex], where t is time in years. Estimate the growth of the tree through the end of the second year by using Simpson’s rule, using two subintervals. (Round the answer to the nearest hundredth.)<\/li>\n
                                9. Given [latex]{\\displaystyle\\int }_{1}^{5}\\left(3{x}^{2}-2x\\right)dx=100[\/latex], approximate the value of this integral using the trapezoidal rule with 16 subdivisions and determine the absolute error.<\/li>\n
                                10. The table represents the coordinates [latex]\\left(x,\\text{ }y\\right)[\/latex] that give the boundary of a lot. The units of measurement are meters. Use the trapezoidal rule to estimate the number of square meters of land that is in this lot.<\/li>\n<\/ol>\n\n\n\n\n\n\n\n\n\n\n
                                  [latex]x[\/latex]<\/th>\n[latex]y[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\n[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
                                  [latex]0[\/latex]<\/td>\n[latex]125[\/latex]<\/td>\n[latex]600[\/latex]<\/td>\n[latex]95[\/latex]<\/td>\n<\/tr>\n
                                  [latex]100[\/latex]<\/td>\n[latex]125[\/latex]<\/td>\n[latex]700[\/latex]<\/td>\n[latex]88[\/latex]<\/td>\n<\/tr>\n
                                  [latex]200[\/latex]<\/td>\n[latex]120[\/latex]<\/td>\n[latex]800[\/latex]<\/td>\n[latex]75[\/latex]<\/td>\n<\/tr>\n
                                  [latex]300[\/latex]<\/td>\n[latex]112[\/latex]<\/td>\n[latex]900[\/latex]<\/td>\n[latex]35[\/latex]<\/td>\n<\/tr>\n
                                  [latex]400[\/latex]<\/td>\n[latex]90[\/latex]<\/td>\n[latex]1000[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n<\/tr>\n
                                  [latex]500[\/latex]<\/td>\n[latex]90[\/latex]<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                    \n
                                  1. The “Simpson” sum is based on the area under a ____.<\/li>\n<\/ol>\n

                                    Error Analysis in Numerical Integration<\/span><\/h2>\n

                                    For the following exercises (1-2), find an upper bound for the error in estimating the given integral using the specified numerical integration method.<\/strong><\/p>\n

                                      \n
                                    1. Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{4}^{5}\\dfrac{1}{{\\left(x - 1\\right)}^{2}}dx[\/latex] using the trapezoidal rule with seven subdivisions.<\/li>\n
                                    2. Find an upper bound for the error in estimating [latex]{\\displaystyle\\int }_{2}^{5}\\dfrac{1}{x - 1}dx[\/latex] using Simpson’s rule with [latex]n=10[\/latex] steps.<\/li>\n<\/ol>\n

                                      For the following exercises (3-4), estimate the minimum number of subintervals needed to approximate the given integral with the specified error tolerance using the trapezoidal rule.<\/strong><\/p>\n

                                        \n
                                      1. Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{1}^{4}\\left(5{x}^{2}+8\\right)dx[\/latex] with an error magnitude of less than [latex]0.0001[\/latex] using the trapezoidal rule.<\/li>\n
                                      2. Estimate the minimum number of subintervals needed to approximate the integral [latex]{\\displaystyle\\int }_{2}^{3}\\left(2{x}^{3}+4x\\right)dx[\/latex] with an error of magnitude less than [latex]0.0001[\/latex] using the trapezoidal rule.<\/li>\n<\/ol>\n

                                        Improper Integrals<\/span><\/h2>\n

                                        For the following exercises (1-4), evaluate the following integrals. If the integral is not convergent, answer “divergent.”<\/strong><\/p>\n

                                          \n
                                        1. [latex]{\\displaystyle\\int }_{2}^{4}\\dfrac{dx}{{\\left(x - 3\\right)}^{2}}[\/latex]<\/li>\n
                                        2. [latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{1}{\\sqrt{4-{x}^{2}}}dx[\/latex]<\/li>\n
                                        3. [latex]{\\displaystyle\\int }_{1}^{\\infty }x{e}^{\\text{-}x}dx[\/latex]<\/li>\n
                                        4. Without integrating, determine whether the integral [latex]{\\displaystyle\\int }_{1}^{\\infty }\\dfrac{1}{\\sqrt{{x}^{3}+1}}dx[\/latex] converges or diverges by comparing the function [latex]f\\left(x\\right)=\\dfrac{1}{\\sqrt{{x}^{3}+1}}[\/latex] with [latex]g\\left(x\\right)=\\dfrac{1}{\\sqrt{{x}^{3}}}[\/latex].<\/li>\n<\/ol>\n

                                          For the following exercises (5-13), determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.<\/strong><\/p>\n

                                            \n
                                          1. [latex]{\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}x}\\cos{x}dx[\/latex]<\/li>\n
                                          2. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{\\text{ln}x}{\\sqrt{x}}dx[\/latex]<\/li>\n
                                          3. [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\dfrac{1}{{x}^{2}+1}dx[\/latex]<\/li>\n
                                          4. [latex]{\\displaystyle\\int }_{-2}^{2}\\dfrac{dx}{{\\left(1+x\\right)}^{2}}[\/latex]<\/li>\n
                                          5. [latex]{\\displaystyle\\int }_{0}^{\\infty }\\sin{x}dx[\/latex]<\/li>\n
                                          6. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dx}{\\sqrt[3]{x}}[\/latex]<\/li>\n
                                          7. [latex]{\\displaystyle\\int }_{-1}^{2}\\dfrac{dx}{{x}^{3}}[\/latex]<\/li>\n
                                          8. [latex]{\\displaystyle\\int }_{0}^{3}\\dfrac{1}{x - 1}dx[\/latex]<\/li>\n
                                          9. [latex]{\\displaystyle\\int }_{3}^{5}\\dfrac{5}{{\\left(x - 4\\right)}^{2}}dx[\/latex]<\/li>\n<\/ol>\n

                                            In the following exercise, determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.<\/strong><\/p>\n

                                              \n
                                            1. [latex]{\\displaystyle\\int }{1}^{\\infty }\\dfrac{dx}{\\sqrt{x}+1}[\/latex]; compare with [latex]{\\displaystyle\\int }{1}^{\\infty }\\dfrac{dx}{2\\sqrt{x}}[\/latex].<\/li>\n<\/ol>\n

                                              For the following exercises (15-19), evaluate the integrals. If the integral diverges, answer “diverges.”<\/strong><\/p>\n

                                                \n
                                              1. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dx}{{x}^{\\pi }}[\/latex]<\/li>\n
                                              2. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dx}{1-x}[\/latex]<\/li>\n
                                              3. [latex]{\\displaystyle\\int }_{-1}^{1}\\dfrac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\n
                                              4. [latex]{\\displaystyle\\int }_{0}^{e}\\text{ln}\\left(x\\right)dx[\/latex]<\/li>\n
                                              5. [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\dfrac{x}{{\\left({x}^{2}+1\\right)}^{2}}dx[\/latex]<\/li>\n<\/ol>\n

                                                For the following exercises (20-27), evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.<\/strong><\/p>\n

                                                  \n
                                                1. [latex]{\\displaystyle\\int }_{0}^{9}\\dfrac{dx}{\\sqrt{9-x}}[\/latex]<\/li>\n
                                                2. [latex]{\\displaystyle\\int }_{0}^{3}\\dfrac{dx}{\\sqrt{9-{x}^{2}}}[\/latex]<\/li>\n
                                                3. [latex]{\\displaystyle\\int }_{0}^{4}x\\text{ln}\\left(4x\\right)dx[\/latex]<\/li>\n<\/ol>\n

                                                  For the following exercises (23-27), solve each problem.<\/strong><\/p>\n

                                                    \n
                                                  1. Evaluate [latex]{\\displaystyle\\int }_{.5}^{1}\\dfrac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]. (Be careful!) (Express your answer using three decimal places.)<\/li>\n
                                                  2. Evaluate [latex]{\\displaystyle\\int }_{2}^{\\infty }\\dfrac{dx}{{\\left({x}^{2}-1\\right)}^{\\dfrac{3}{2}}}[\/latex].<\/li>\n
                                                  3. Find the area of the region bounded by the curve [latex]y=\\dfrac{7}{{x}^{2}}[\/latex], the x-axis, and on the left by [latex]x=1[\/latex].<\/li>\n
                                                  4. Find the area under [latex]y=\\dfrac{5}{1+{x}^{2}}[\/latex] in the first quadrant.<\/li>\n
                                                  5. Find the volume of the solid generated by revolving about the y-axis the region under the curve [latex]y=6{e}^{-2x}[\/latex] in the first quadrant.<\/li>\n<\/ol>\n

                                                    The Laplace transform of a continuous function over the interval [latex]\\left[0,\\infty \\right)[\/latex] is defined by [latex]F\\left(s\\right)={\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}sx}f\\left(x\\right)dx[\/latex]. This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of [latex]F[\/latex] is the set of all real numbers [latex]s[\/latex] such that the improper integral converges. Find the Laplace transform [latex]F[\/latex] of each of the following functions (28-30) and give the domain of [latex]F[\/latex].<\/strong><\/p>\n

                                                      \n
                                                    1. [latex]f\\left(x\\right)=1[\/latex]<\/li>\n
                                                    2. [latex]f\\left(x\\right)=\\cos\\left(2x\\right)[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":667,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/785"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/785\/revisions"}],"predecessor-version":[{"id":1743,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/785\/revisions\/1743"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/667"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/785\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=785"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=785"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=785"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=785"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}