When we're integrating a non-negative function from *a* to *b* where a**x-axis 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.

## Practice:

Let *f* (*x*) = 2*x*. Find . | |

When we look at the graph, we see a big triangle below the *x*-axis and a little triangle above the *x*-axis. The total area above the *x*-axis and below *f* is The total area below the *x*-axis and above *f* is To get the integral of *f*, we take the area above the *x*-axis and subtract the area below the *x*-axis:
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Find | |

This graph consists of two triangles, both below the *x*-axis: The total area between the *x* axis and -|*x*| above the axis is 0. The total area between the *x*-axis and -|*x*| below the axis is 20. So
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If , what is *a*? | |

Draw a picture. We're integrating 2*x* from *a* to 1. Since the integral value is negative there must be some area below the *x*-axis, so we can assume *a* is negative. This line forms two triangles. We know the dimensions of the one above the *x*-axis: The area of the triangle above the *x*-axis is 1. For the triangle below the *x*-axis, we can say what its dimensions are in terms of *a*. Since *a* is negative, the base of the triangle is the positive distance -a. Similarly, the height of the triangle is -2*a*. The area of this triangle is We now have This is a perfectly reasonable equation that we can solve for *a*. Since we know *a* needs to be negative, *a* = -3 is the answer. | |

Find the integral.

where* *

Answer

- This function, graphed on [-5,2], produces two rectangles:

We count the little rectangle (square) above the *x*-axis positively, and the big one below the *x*-axis negatively:

Find the integral.

Answer

- Now we have two triangles:

First we need to figure out where the line hits the *x*-axis, since that will tell us what the bases of the triangles are.Since

when *x* = 4, we know *x* = 4 is the point in between the two triangles.

Now we can figure out the necessary dimensions:

And find the integral:

Find the integral. (hint: draw pictures)

Hint

don't try to calculate any areas

Answer

- The hint says to not try to calculate any areas. Since the sin function is odd, the area below the
*x*-axis and the area above the *x*-axis are equal:

When we count one area positively and one area negatively, the weighted areas cancel out. So

Find the integral. (hint: draw pictures)

Hint

calculate the missing area

Answer

describes half of a circle sitting on top of the *x*-axis. The area we want is the area inside the purple 2-by-1 rectangle, but not inside the half-circle:

The rectangle has area 2 and the half-circle has area , so

For a sanity check, this is approximately 0.429, which is positive, as would be expected when finding the integral of a non-negative function.

Find the integral.

where *g*(*t*) = 3*t* + 4.

Answer

- Here we have another triangle:

- If and
*a* is positive, what is *a*?

Answer

- This graph consists of two equally sized triangles:

The triangle on the right has area . Since the triangles have the same size, together they have area

So we know . We're given , so 49 and *a*^{2} must be the same. That means *a* = 7.

- If and
*a* is positive, find *a*.

Answer

- This is like the previous problem, only easier! From a graph we can see that the region between the graph of
*x* and the *x*-axis is a triangle with area .

So

which means

9 = *a*^{2}

*a *= 3.

We only keep the answer *a* = 3 since we were told *a* was positive.

- Find
*c* given that .

Answer

- Since the integral is positive, we can assume that
*c* is positive. If we graph the constant *c* on [-10,10] we get a rectangle of height *c* and length 20. This means the area of the rectangle is 20*c*. This must equal 40, which we were told was the value of the integral, so *c* = 2.