Numerical and Improper Integration: Get Stronger

Numerical Integration Methods

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.)

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

In the following exercises (5-9), approximate the integral to three decimal places using the indicated rule.

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

For the following exercises (10-20), solve each problem.

  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.
  2. Use the trapezoidal rule with four subdivisions to estimate [latex]{\displaystyle\int }_{0}^{0.8}{x}^{3}dx[/latex] to four decimal places.
  3. Using Simpson’s rule with four subdivisions, find [latex]{\displaystyle\int }_{0}^{\dfrac{\pi}{2}}\cos\left(x\right)dx[/latex].
  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.
  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].
  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].
  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.
  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.)
  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.
  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.
[latex]x[/latex] [latex]y[/latex] [latex]x[/latex] [latex]y[/latex]
[latex]0[/latex] [latex]125[/latex] [latex]600[/latex] [latex]95[/latex]
[latex]100[/latex] [latex]125[/latex] [latex]700[/latex] [latex]88[/latex]
[latex]200[/latex] [latex]120[/latex] [latex]800[/latex] [latex]75[/latex]
[latex]300[/latex] [latex]112[/latex] [latex]900[/latex] [latex]35[/latex]
[latex]400[/latex] [latex]90[/latex] [latex]1000[/latex] [latex]0[/latex]
[latex]500[/latex] [latex]90[/latex]
  1. The “Simpson” sum is based on the area under a ____.

Error Analysis in Numerical Integration

For the following exercises (1-2), find an upper bound for the error in estimating the given integral using the specified numerical integration method.

  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.
  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.

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.

  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.
  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.

Improper Integrals

For the following exercises (1-4), evaluate the following integrals. If the integral is not convergent, answer “divergent.”

  1. [latex]{\displaystyle\int }_{2}^{4}\dfrac{dx}{{\left(x - 3\right)}^{2}}[/latex]
  2. [latex]{\displaystyle\int }_{0}^{2}\dfrac{1}{\sqrt{4-{x}^{2}}}dx[/latex]
  3. [latex]{\displaystyle\int }_{1}^{\infty }x{e}^{\text{-}x}dx[/latex]
  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].

For the following exercises (5-13), determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.

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

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.

  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].

For the following exercises (15-19), evaluate the integrals. If the integral diverges, answer “diverges.”

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

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.

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

For the following exercises (23-27), solve each problem.

  1. Evaluate [latex]{\displaystyle\int }_{.5}^{1}\dfrac{dx}{\sqrt{1-{x}^{2}}}[/latex]. (Be careful!) (Express your answer using three decimal places.)
  2. Evaluate [latex]{\displaystyle\int }_{2}^{\infty }\dfrac{dx}{{\left({x}^{2}-1\right)}^{\dfrac{3}{2}}}[/latex].
  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].
  4. Find the area under [latex]y=\dfrac{5}{1+{x}^{2}}[/latex] in the first quadrant.
  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.

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].

  1. [latex]f\left(x\right)=1[/latex]
  2. [latex]f\left(x\right)=\cos\left(2x\right)[/latex]