Download a PDF of this page here.
Download the Spanish version here.
Essential Concepts
Approximating Areas
- The use of sigma (summation) notation of the form n∑i=1ai is useful for expressing long sums of values in compact form.
- For a continuous function defined over an interval [a,b], the process of dividing the interval into n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
- The width of each rectangle is Δx=b−an
- Riemann sums are expressions of the form n∑i=1f(x∗i)Δx, and can be used to estimate the area under the curve y=f(x). Left- and right-endpoint approximations are special kinds of Riemann sums where the values of {x∗i} are chosen to be the left or right endpoints of the subintervals, respectively.
- Riemann sums allow for much flexibility in choosing the set of points {x∗i} at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.
The Definite Integral
- The definite integral can be used to calculate net signed area, which is the area above the x-axis minus the area below the x-axis. Net signed area can be positive, negative, or zero.
- The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
- Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
- The properties of definite integrals can be used to evaluate integrals.
- The area under the curve of many functions can be calculated using geometric formulas.
- The average value of a function can be calculated using definite integrals.
The Fundamental Theorem of Calculus
- The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value c such that f(c) equals the average value of the function. See the Mean Value Theorem for Integrals.
- The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See the Fundamental Theorem of Calculus, Part 1.
- The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. See the Fundamental Theorem of Calculus, Part 2.
Integration Formulas and the Net Change Theorem
- The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero.
- The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.
Key Equations
- Properties of Sigma Notation
nΣi=1c=nc
nΣi=1cai=cnΣi=1ai
nΣi=1(ai+bi)=nΣi=1ai+nΣi=1bi
nΣi=1(ai−bi)=nΣi=1ai−nΣi=1bi
nΣi=1ai=mΣi=1ai+nΣi=m+1ai - Sums and Powers of Integers
nΣi=1i=1+2+⋯+n=n(n+1)2
nΣi=1i2=12+22+⋯+n2=n(n+1)(2n+1)6
nΣi=1i3=13+23+⋯+n3=n2(n+1)24 - Left-Endpoint Approximation
A≈Ln=f(x0)Δx+f(x1)Δx+⋯+f(xn−1)Δx=nΣi=1f(xi−1)Δx - Right-Endpoint Approximation
A≈Rn=f(x1)Δx+f(x2)Δx+⋯+f(xn)Δx=nΣi=1f(xi)Δx - Definite Integral
∫baf(x)dx=limn→∞nΣi=1f(x∗i)Δx - Properties of the Definite Integral
∫aaf(x)dx=0
∫abf(x)dx=−∫baf(x)dx
∫ba[f(x)+g(x)]dx=∫baf(x)dx+∫bag(x)dx
∫ba[f(x)−g(x)]dx=∫baf(x)dx−∫bag(x)dx
∫bacf(x)dx=c∫baf(x)dx for constant c
∫baf(x)dx=∫caf(x)dx+∫bcf(x)dx - Mean Value Theorem for Integrals
If f(x) is continuous over an interval [a,b], then there is at least one point c∈[a,b] such that f(c)=1b−a∫baf(x)dx. - Fundamental Theorem of Calculus Part 1
If f(x) is continuous over an interval [a,b], and the function F(x) is defined by F(x)=∫xaf(t)dt, then F′(x)=f(x). - Fundamental Theorem of Calculus Part 2
If f is continuous over the interval [a,b] and F(x) is any antiderivative of f(x), then ∫baf(x)dx=F(b)−F(a). - Net Change Theorem
F(b)=F(a)+∫baF'(x)dx or ∫baF'(x)dx=F(b)−F(a)
Glossary
- average value of a function
- (or fave) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval
- definite integral
- a primary operation of calculus; the area between the curve and the x-axis over a given interval is a definite integral
- integrable function
- a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as n goes to infinity exists
- integrand
- the function to the right of the integration symbol; the integrand includes the function being integrated
- left-endpoint approximation
- an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle
- limits of integration
- these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated
- lower sum
- a sum obtained by using the minimum value of f(x) on each subinterval
- net change theorem
- if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity
- net signed area
- the area between a function and the x-axis such that the area below the x-axis is subtracted from the area above the x-axis; the result is the same as the definite integral of the function
- partition
- a set of points that divides an interval into subintervals
- regular partition
- a partition in which the subintervals all have the same width
- riemann sum
- an estimate of the area under the curve of the form A≈nΣi=1f(x∗i)Δx
- right-endpoint approximation
- the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle
- sigma notation
- (also, summation notation) the Greek letter sigma (Σ) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it
- total area
- total area between a function and the x-axis is calculated by adding the area above the x-axis and the area below the x-axis; the result is the same as the definite integral of the absolute value of the function
- upper sum
- a sum obtained by using the maximum value of f(x) on each subinterval
- variable of integration
- indicates which variable you are integrating with respect to; if it is x, then the function in the integrand is followed by dx
Study Tips
Approximating Area
- Practice identifying left and right endpoints for given intervals. Compare left and right approximations for various functions
- Visualize how increasing n affects the approximation
- Remember that for increasing functions, Ln≤ True Area ≤Rn
- For decreasing functions, the inequality is reversed
Riemann Sums
- Practice identifying whether a sum is an upper or lower Riemann sum
- For increasing functions, right endpoints give upper sums, left endpoints give lower sums
- For decreasing functions, left endpoints give upper sums, right endpoints give lower sums
- Remember that upper sums overestimate and lower sums underestimate the true area
- Practice calculating Riemann sums with different choices of x∗i
Defining and Evaluating Definite Integrals
- Practice setting up Riemann sums for various functions and intervals.
- Understand the difference between definite and indefinite integrals.
- Familiarize yourself with basic geometric formulas (e.g., areas of rectangles, triangles, circles) for simple integral evaluations.
- Visualize the definite integral as the signed area under a curve.
- Pay attention to the limits of integration and ensure they’re in the correct order.
- Remember that the variable of integration is a dummy variable – changing it doesn’t affect the result.
Area and the Definite Integral
- Practice sketching graphs and identifying areas above and below the x-axis.
- When solving problems, decide whether you need net signed area or total area based on the context.
- For displacement problems, remember:
- Positive velocity: Moving in positive direction
- Negative velocity: Moving in negative direction
- For more complex functions, be prepared to break the interval into subintervals where the function is consistently above or below the x-axis.
- Remember that the definite integral can be interpreted as a sum of signed areas.
Properties of the Definite Integral
- Memorize the basic properties of definite integrals and practice applying them to simplify complex integrals.
- When comparing integrals, always check the interval of integration and the relative positions of the functions’ graphs.
- Remember that the Comparison Theorem only applies when a≤b.
- Practice breaking down complex integrals into simpler parts using the sum, difference, and interval splitting properties.
- When using the interval splitting property, remember it works for any value of c, not just those between a and b.
Average Value of a Function
- Visualize the average value as the height of an equivalent rectangle.
- Remember that the average value formula always includes the factor 1b−a outside the integral.
- For simple functions, try to evaluate the integral geometrically before using antiderivatives.
- Be careful with units: the average value has the same units as the function f(x).
- When possible, use symmetry to simplify calculations (e.g., for even/odd functions on symmetric intervals).
The Mean Value Theorem for Integrals
- Visualize the theorem: imagine a horizontal line cutting through the function graph such that areas above and below the line are equal.
- Remember that c is not necessarily unique; there may be multiple points satisfying the theorem.
- Use this theorem to estimate integrals or function values without direct computation.
- When solving problems, always check if c is within the given interval [a,b].
Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives
- Understand the difference between F(x) and a definite integral with fixed limits.
- Visualize F(x) as an accumulation function, tracking the area under f(t) as x varies.
- Remember that the theorem requires f(x) to be continuous on the interval.
- When dealing with variable limits of integration, use the chain rule in conjunction with the theorem.
Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem
- Remember to evaluate the antiderivative at both endpoints and subtract. Be careful with signs when evaluating at the endpoints.
- Don’t forget to include the negative sign when the lower limit is larger than the upper limit.
- Connect the result of the definite integral to the area under the curve, but remember they’re not always the same (consider negative areas).
The Net Change Theorem
- Understand the difference between net change and total change.
- Remember that the integral of velocity gives displacement, not necessarily distance traveled.
- For total distance, integrate the absolute value of the velocity function.
- Be careful with signs when the rate of change switches from positive to negative or vice versa.
- Practice breaking down problems into subintervals when the rate function changes behavior.
Integrating Even and Odd Functions
- Practice identifying even and odd functions algebraically and graphically.
- For even functions, remember you can halve the interval and double the result.
- For odd functions, visualize the cancellation of areas above and below the x-axis.
- Remember that not all functions are even or odd; some are neither.
- Practice decomposing functions into even and odd parts.