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Section 5.4 Theorems About Definite Integrals

Explore the fundamental properties of definite integrals, including limits of integration and the effects of continuous functions. Learn how to evaluate integrals using symmetry by identifying even and odd functions and their unique characteristics. This guide covers the area between two curves, explaining how to find exact areas under graphs through definite integrals. Step-by-step examples illustrate the process of comparing functions and calculating areas enclosed by curves, such as y = e^x + 1 and y = x, within specified limits.

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Section 5.4 Theorems About Definite Integrals

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  1. Section 5.4Theorems About Definite Integrals

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

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

  4. Example • Given that find the following:

  5. 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

  6. EXAMPLE Given that Find

  7. Comparison of Definite Integrals • Let f and g be continuous functions

  8. Example • Explain why

  9. 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!

  10. Find the exact value of the area between the graphs of y = e x + 1 and y = xfor0 ≤ x ≤ 2

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

  12. Now let’s add in the graph of y = x

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

  14. 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

  15. 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

  16. Find the exact value of the area between the graphs of y = x + 1 and y = 7 - x for0 ≤ x ≤ 4

  17. Let’s shade in the area we are looking for

  18. 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

  19. 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

  20. Find the exact value of the area enclosed by the graphs of y = x2 and y = 2 - x2

  21. Let’s shade in the area we are looking for

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

  23. 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

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