Write a definite integral that represents the area of the region. (Do not evaluate the integral.)

Question

y₁=x²-3x

y₂=0

Answer 1

The definite integral is the area under the curve of y₁ = x² - 3x from the place the curve first intersects y₂ = 0 to the second place it intersects y₂ = 0. The curve looks like this:

To find the place they intersect, we set them equal to each other:

x² - 3x = 0

x(x - 3) = 0

x = 0 and x = 3.

So the integral becomes:

Since the area is below the x-axis (see graph), the result will turn out to be negative. So, since we are being asked to find the area of the region, I'd say we want to take the absolute value of this integral to get a positive answer for the area of the region. Just put absolute value signs around the integral.