- Topics At a Glance
- Left-Hand Sum
- Right-Hand Sum
- Comparing Right- and Left-Hand Sums
- Error in Left- and Right-Hand Sums
- Midpoint Sum
- Midpoint Sums with Shortcuts
- Over or Under Estimates
- Trapezoid Sum
- Trapezoid Sum with Shortcuts
- Over or Under Estimates
- Comparison of Sums
- Definite Integrals of Non-Negative Functions
- Definite Integrals of Real-Valued Functions
- Conditions for Integration
- General Riemann Sums
**Properties of Definite Integrals****Single-Function Properties**- Talking About Two Functions
- Thinking Backwards
- Average Value
- Averages with Numbers
- Averages with Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

These properties require a little more explanation. We're still assuming *f* is an integrable function.

- Let
*c*be a constant. Then

As an example, let *f* = *x* on [0,*b*] and let *c* = 3.

When we stretch *f* by 3 we don't change the base of the triangle, but we do stretch the height by 3. We also multiply the area by 3.

When we rewrite

as

we say we're "pulling out the constant" or "pulling the constant out of the integral.

- We can switch the limits of integration if we also switch the sign:

We've been focusing on integrals from *a* to *b* where a**b**** to a. If f is non-negative then as we accumulate area from left to right, we weight the area positively. As we accumulate area from right to left, we weight it negatively.**

This is easiest to see when a**f**** on [ a,b] is the weighted area between f and the x-axis on [a,b]. The integral of f on [b,c] is the weighted area between f and the x-axis on [b,c]. When we add these weighted areas, we get the weighted area between f and the x-axis on [a,c], which is the integral of f on [a,c]. When a,b,c don't fall nicely in order, this is still true.**

- If
*f*(*x*) is an even function then

If *f* is even then it's reflected across the *y*-axis.This means whatever area is on one side of the graph is also on the other side of the graph. The weighted area between *f* and the *x*-axis on [-*a,a*] is then double the weighted area between *f* and the *x*-axis on [0,*a*].

- If
*f*is an odd function then

You can see that the area between *f* and the horizontal axis is the same on both sides, but on one side the area is above the *x*-axis and on the other it's below the *x*-axis. Odd functions will always work this way. If we integrate from -a to *a* the weighted areas above and below the *x*-axis will cancel each other out.

Example 1

If , what is |

Example 2

If , what is |

Example 3

The function
What is ? |

Example 4

Let |

Example 5

If , what is ? |

Exercise 1

Assuming that

evaluate the following expression.

Exercise 2

Assuming that

evaluate the following expression.

Exercise 3

Assuming that

and that *f* (*x*) is an even function,

find the following integral.

Exercise 4

Assuming that

and that *f* ( *x* ) is an even function,

find the following integral.

- .

Exercise 5

Assuming that

and that *f* is an odd function,

what is