Definite Integrals of Real-Valued Functions - At A Glance

Hint

Draw pictures.

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:

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:

Answer

Since the sine 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

Answer

The function

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.

Hint

draw pictures

Answer

Here we have another triangle:

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 a² must be the same. That means a = 7.

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 = a²

a = 3.

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

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 20c. This must equal 40, which we were told was the value of the integral, so c = 2.