The Fundamental Theorem of Calculus Objective: The use the Fundamental Theorem of Calculus to evaluate Definite Integrals.

The Fundamental Theorem of Calculus Pt. 1 If f is continuous on [a, b] and F is any antiderivative of f on [a, b], then This is also expressed in this form.

Example 1 Evaluate

Example 1 Evaluate

Example 2 Evaluate

Example 2 Evaluate

Example 3 a)Find the area under the curve y = cosx over the interval [0,  /2]

Example 3 a)Find the area under the curve y = cosx over the interval [0,  /2] Since the cosx > 0 over the interval [0,  /2] the area under the curve is

Example 3 b) Make a conjecture about the value of the integral

Example 3 b) Make a conjecture about the value of the integral The given integral can be interpreted as the signed area between the graph of y = cosx and the interval [0,  ]. We can see that the area above and below the x-axis is the same. The value is zero.

Example 4 Evaluate

Example 4 Evaluate

Example 5 Here are a few more examples.

Example 5 Here are a few more examples.

Example 6 Remember the two theorems we had earlier.

Example 7 Evaluate if

Example 7 Evaluate if

Total Area If f is a continuous function on the interval [a, b], then we define the total area between the curve y = f(x) and the interval [a, b] to be

Total Area To compute total area, begin by dividing the interval of integration into subintervals on which f(x) does not change sign. On the subintervals on which 0 < f(x) replace |f(x)| with f(x), and on the subintervals for which f(x) < 0 replace |f(x)| with –f(x). Adding the resulting integrals will give total area.

Example 8 Find the total area between the curve y = 1 – x 2 and the x-axis over the interval [0, 2].

Example 8 Find the total area between the curve y = 1 – x 2 and the x-axis over the interval [0, 2]. The area is given by

Dummy Variables The variable used in integration is not important. As long as everything is in terms of one variable, it can be anything you want. Because the variable of integration in a definite integral plays no role in the end result, it is often referred to as a dummy variable. You can change the variable of integration without changing the value of the integral.

The Mean-Value Theorem for Integrals Let f be a continuous function on [a, b], and let m and M be the minimum and maximum values of f(x) on this interval. Consider the rectangles of heights m and M over the interval [a, b]. It is clear geometrically that the area under f(x) is at least as large as the rectangle of height m and no larger than the area of the rectangle of height M.

The Mean-Value Theorem for Integrals Theorem If f is continuous on a closed interval [a, b], then there is at least one point x * in [a, b] such that

Example 9 Since f(x) = x 2 is continuous on the interval [1, 4], the Mean-Value Theorem for Integrals guarantees that there is a point x * in [1, 4] such that

Theorem The Fundamental Theorem of Calculus, Part 2 If f is continuous on an interval I, then f has an antiderivative on I. In particular if a is any point in I, then the function f defined by is an antiderivative of f on I; that is, for each x in I, or in an alternative notation

Example 10 Find

Example 10 Find

Example 10 Find Let’s integrate and the take the derivative to confirm this result.

Chain Rule The Fundamental Theorem of Calculus, Part 2 also has a form the applies the chain rule.

Chain Rule Let’s integrate, then take the derivative to confirm this rule.

Integrating Rates of Change The Fundamental Theorem of Calculus has a useful interpretation that can be seen by rewriting it in a slightly different form. Since F is an antiderivative of f on the interval [a, b], we can use the relationship F / (x) = f(x) to rewrite it like this:

Integrating Rates of Change In this formula we can view F / (x) as the rate of change of F(x) with respect to x, and we can view F(b) – F(a) as the change in the value of F(x) as x increases from a to b. This leads us to the following principle Integrating Rate of Change- Integrating the rate of change of F(x) with respect to x over an interval [a, b] produces the amount of change in the value of F(x) that occurs as x increases from a to b.

Integrating Rates of Change Here are some examples of this idea: 1.If P(t) is a population at time t, the P / (t) is the rate at which the population is changing at time t and is the amount of change in the population between times t 1 and t 2.

Integrating Rates of Change Here are some examples of this idea: 2.If A(t) is the area of an oil spill at time t, and A / (t) is the rate at which the area of the spill is changing at time t, and is the amount of change in the area of the spill between times t 1 and t 2.

Homework Section odd 59