Question

AP Calc pls help

Find the area of the region bounded by the graphs of y = x2 − 4x and y = x − 4.

Answer 1

9/2

Explanation:

First, you want to graph these two functions and find where they intersect so you can figure out the lower and upper bounds you need on your integral. I just entered it in on my TI-84 and had it figure out the intersections for me: (1, -3) and (4, 0). You could also set these two equations equal to one another and figure out their intersection point algebraically.

This means your integral should have an upper bound of 4 and a lower bound of 1, since I want to integrate with respect to x. You could also integrate with respect to y; it depends on your preference. Since the equations are already solved for y, it's easier to integrate with respect to x. To find the area between two curves, you want to integrate the height of the space between these two curves, so what you do is the curve on top minus the curve on the bottom. In our case, x - 4 is the above the area between the curve, and x^2 - 4x is below the area between the curve. So the integral looks like this:

`\int\limits^4_1 {(x - 4) - (x^2 - 4x)} \, dx`

My result is that the area is 9/2, which you can find by simplifying the integral. I'd be happy to do it for you, but using the formula thing on this site is kind of a huge pain. It'll take me a second to write down the work if you need it, but hopefully just seeing how the integral is set up will be enough to get you there!