: At a Glance

Introduction to :

"Thinking backwards" shows up in a lot of places. That means it must be important! Here, "thinking backwards" means instead of using a known integral value to evaluate an expression, we'll work backwards from an equation to find the integral value.

Think of the integral you're looking for as a unit. Instead of solving for x, you'll be solving for something like . If your equation contains a slightly different integral than the one you want, use the properties to manipulate the integral until it's what you're looking for.

Example 1

Find given that

Example 2

Find given that

Example 3

Combine the tricks used in the previous two examples to find

given that

Example 4

Find assuming that f is even and that

Example 5

Find assuming that f is even and that

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Exercise 1

Find given that

Exercise 2

Find given that

Exercise 3

Find given that

Exercise 4

Find given that f (x) is even and that
Answer
Split up the integral, then evaluate the integral of the constant (don't be fooled by the order of the limits):
Since we know this whole mess must equal 34,
Switch the limits and switch the signs, then use the fact that f is even:
We conclude that .

Exercise 5

Find given that f is odd and
Answer
Split up the integral, pull out the constant, and evaluate the integral of 4:
Since this needs to equal 26, we'll clean some things up.
Now switch the limits and switch the sign:
Split up the integral:
Since f is odd, its integral from -2 to 2 is zero, so

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