- Topics At a Glance
- Integrating the Velocity Function
- Negative Velocity
- Change in Position
- Total Distance Travelled vs. Change in Position
- About the Fundamental Theorem of Calculus (FTC)
**Using the FTC to Evaluate Integrals****Integrating with Letters**- Order of Limits of Integration
- Why the Choice of Antiderivative Doesn't Matter
- Average Values
- Units
- Word Problems
- The Second Fundamental Theorem of Calculus
- Letters
- Antiderivatives
- Finding Derivatives
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

No, we're not talking about snail mail. We're throwing

If we have a constant instead of a number for a limit of integration, not much changes. We apply the FTC, and write a constant instead of a number where it's appropriate to do so.

If *c* is a constant greater than 0, find .

Answer.

We apply the FTC like always, but use *c* for the upper limit of integration instead of a number.

Now we can answer questions like this:

If , what is *c*?

Answer.

We just worked out that

So if the integral is equal to 2, it means

Solving, we get

*c*^{3} = 6

When there's a constant in the integrand, you have to take it into account while finding the antiderivative. If there's a constant in the integrand, that constant will also show up in the antiderivative.

A word of warning: when there are constants in the integrand, it can be easy to get mixed up when it comes time to put in the limits of integration. Do they plug into the *a* or the *x*? The answer is that the limits of integration plug into the variable of integration–in this example, the limits of integration (1 and 5) went into the variable of integration *x*, not into the constant *a*.

**Be Careful:** Plug the limits of integration into the variable of integration. If there are constants in the integrand, leave those alone–do NOT plug the limits of integration into any constants.

Example 1

Find where |

Example 2

Let |

Example 3

Find if (a) (b) (c) |

Exercise 1

Find the integral.

where *a* > 2 is a constant

Exercise 2

Find the integral.

where *a* and *c* are constants, *a* < *c*

Exercise 3

Find the integral.

, where *a* is a nonzero constant

Exercise 4

Find the integral.

where *u* is a nonzero constant

Exercise 5

Find the integral.

where *a* is a constant greater than zero

Exercise 6

Find the integral.

where *C* and D are constants, *C* < *D*

Exercise 7

Find the integral.

where *a* is a nonzero constant

Exercise 8

Find the integral.

where *a* is a constant greater than zero

Exercise 9

Find the integral.

where *b* > 1 and *a* ≠ 0 are constants