Find the area between curves

Question

y = 12 - x, y = √x, and y = 1

Determine its entire area in units ²

Area =

Answer 1

Sketch it out.

You can see that the answer lies in the shaded region in the middle.

So, find the intersection points.

The intersection of red (y = √x) and blue (y = 1) is found by setting them equal:

√x = 1 or x = 1 meaning the intersection point is (1, 1)

The intersection of red (y = √x) and green (y = 12 - x) is found by setting them equal:

√x = 12 - x

(√x)² = (12 - x)²

x = 144 - 24x + x²

x² -25x + 144 = 0

(x - 9)(x - 16) = 0

x = 9 and x = 16 but plugging these back in, we see that only x = 9 is a solution.

Plug x = 9 back in to find y = 3 so the intersection point is (9, 3)

The intersection of green (y = 12 - x) and blue (y = 1) is found by setting them equal:

12 - x = 1 or x = 11 meaning the intersection point is (11, 1)

Now, set up your integrals:

A = ∫(red - blue)dx from x = 1 to 9 + ∫(green - blue)dx from x = 9 to 11

Then plug in the functions red is "√x", blue is "1", and green is "12 - x" and perform the integrations. You can use a calculator to do the integrations if your teacher allows or just take the antiderivatives and use the Fundamental Theorem.

Answer 2

The area defined by y=1 and y=12-x and y=sqrt(x) can be divided into two parts. The first part is 1<=x<=9 and the second is 9<x<=11. The area could be found by finding the integral from 1 to 9 of sqrt(x)-1 and adding that result to the integral from 9 to 11 of 12-x-1 yielding 11.333