Section 5.4 Theorems About Definite Integrals

Properties of Limits of Integration If a, b, and c are any numbers and f is a continuous function, then

Properties of Sums and Constant Multiples of the Integrand Let f and g be continuous functions and let c be a constant, then

Example Given that find the following:

Using Symmetry to Evaluate Integrals An EVEN function is symmetric about the y-axis An ODD function is symmetric about the origin If f is EVEN, then If f is ODD, then

EXAMPLE Given that Find

Comparison of Definite Integrals Let f and g be continuous functions

Example Explain why

The Area Between Two Curves If the graph of f(x) lies above the graph of g(x) on [a,b], then Area between f and g on [a,b] Let’s see why this works!

Find the exact value of the area between the graphs of y = e x + 1 and y = x for 0 ≤ x ≤ 2

This is the graph of y = e x + 1 What does the integral from 0 to 2 give us?

Now let’s add in the graph of y = x

Now the integral of x from 0 to 2 will give us the area under x

So if we take the area under e x + 1 and subtract out the area under x, we get the area between the 2 curves

So we find the exact value of the area between the graphs of y = e x + 1 and y = x for 0 ≤ x ≤ 2 with the integral Notice that it is the function that was on top minus the function that was on bottom

Find the exact value of the area between the graphs of y = x + 1 and y = 7 - x for 0 ≤ x ≤ 4

Let’s shade in the area we are looking for

Notice that these graphs switch top and bottom at their intersection Thus we must split of the integral at the intersection point and switch the order

So to find the exact value of the area between the graphs of y = x + 1 and y = 7 - x for 0 ≤ x ≤ 4 we can use the following integral

Find the exact value of the area enclosed by the graphs of y = x 2 and y = 2 - x 2

Let’s shade in the area we are looking for

In this case we weren’t given limits of integration Since they enclose an area, we use their intersection points for the limits

So to find the exact value of the area enclosed by the graphs of y = x 2 and y = 2 - x 2 we can use the following integral