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Definite Integrals

Definite Integrals

At a Glance - Definite Integrals of Real-Valued Functions

When we're integrating a non-negative function from a to b, the integral can be thought of as the "area under the curve" of the function. However, most of the time we can't count on having a non-negative function to integrate.

Assume f is a function that's allowed to take on negative values, and we're integrating from a to b with a < b. Then  is the weighted sum of the areas between the graph of f and the x-axis. We look at all areas between f and the x-axis. If they're on top of the x-axis we count them positively. If they're below the x-axis we count them negatively.

In other words, we add all the areas on top of the x-axis, then subtract all the areas below the x-axis.

Example 1

Let f(x) = 2x. Find .


Example 2

Find 


Example 3

If , what is a?


Exercise 1

Find the integral.

where 


Exercise 2

Find the integral.


Exercise 3

Find the integral.


Exercise 4

Find the integral.


Exercise 5

Find the integral. 

where g(t) = 3t + 4.


Exercise 6

  • If  and a is positive, what is a?

Exercise 7

  • If  and a is positive, find a.

Exercise 8

  • Find c given that .

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