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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.
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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 = xfor0 ≤ x ≤ 2
This is the graph of y = e x + 1 What does the integral from 0 to 2 give us?
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 = xfor0 ≤ x ≤ 2with 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 for0 ≤ x ≤ 4
Thus we must split of the integral at the intersection point and switch the order Notice that these graphs switch top and bottom at their intersection
So to find the exact value of the area between the graphs of y = x + 1 and y = 7 - x for0 ≤ x ≤ 4we can use the following integral
Find the exact value of the area enclosed by the graphs of y = x2 and y = 2 - x2
Since they enclose an area, we use their intersection points for the limits In this case we weren’t given limits of integration
So to find the exact value of the area enclosed by the graphs of y = x2 and y = 2 - x2 we can use the following integral