### Calculus meets Functions

Package of problems

## Things you might have noticed

Which of the integrals below can be evaluated if we know that

$\int_0^3 f(x) \ dx = 7 \ \text{and} \ \int_0^3 g(x)\ dx =4\text{?}$

How are the functions we are being asked to integrate related to $f(x)$ and $g(x)$?

Are the limits of these new integrals the same as the original integrals, or are they related to the original limits in a way that we can make use of?

As you read, you may find it helpful to sketch some possible graphs of the functions involved. You could consider some particular examples that you’re comfortable working with, or try to work with more general sketch graphs of $f(x)$ and $g(x).$

For the remaining integrals we need more information, but we could try to use the additional pieces of information to help us.

Which of the integrals can be evaluated if we have the following extra information?

1. $f(x)$ is symmetric about $x=3$
1. $g(x)$ is an odd function
1. $f(x)$ is an even function