Why is the upper limit 4 and not 16?

Question

Find the area enclosed by f(y)=y/4 g(y)=y^1/2

My teacher wrote the integral from 0 to 4 why wouldn't it be 16 set them equal to each other and 4=4

Answer 1

The first equation is:

x = y/4

or y = 4x

The second equation is:

x = √y

or y = x², x ≥ 0

The intersection is (4,16)

The area in terms of x integral is:

∫ (4x - x²) dx, with lower limit 0 and upper limit 4

However, the area in terms of y integral is:

∫ (√y - y/4) dy, with lower limit 0 and upper limit 16

So, your professor is correct if he has set-up the integral in terms of x. Both integrals will give you the same answer.