5: Investigating Integrals
- Page ID
- 116604
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- 5.1: Antiderivatives and Indefinite Integrals
- At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function f , how do we find a function with the derivative f and why would we be interested in such a function?
- 5.2: Approximating Areas
- In this section, we develop techniques to approximate the area between a curve, defined by a function f(x), and the x-axis on a closed interval [a,b]. Like Archimedes, we first approximate the area under the curve using shapes of known area (namely, rectangles). By using smaller and smaller rectangles, we get closer and closer approximations to the area. Taking a limit allows us to calculate the exact area under the curve.
- 5.3: The Definite Integral
- If f(x) is a function defined on an interval [a,b], the definite integral of f from a to b is given by \[∫^b_af(x)dx=\lim_{n→∞} \sum_{i=1}^nf(x^∗_i)Δx,\] provided the limit exists. If this limit exists, the function f(x) is said to be integrable on [a,b], or is an integrable function. The numbers a and b are called the limits of integration; specifically, a is the lower limit and b is the upper limit. The function f(x) is the integrand, and x is the variable of integration.
- 5.4: The Fundamental Theorem of Calculus
- The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this is not always easy.
- 5.5: The Net Change Theorem, Distances, and Symmetry
- 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.
- 5.6: Integration by Substitution
- In this section we examine a technique, called integration by substitution, to help us find antiderivatives. Specifically, this method helps us find antiderivatives when the integrand is the result of a chain-rule derivative.