find the area of the shaded region

Question

Answer 1

It is done the same way as the method you are probably familiar with,

but with the roles of x and y revered.

Namely integrate the difference between the two functions *over y*.

First let's do the indefinite integral

F(y) = ∫(3y - y² -(y² - 4y))dy

= ∫(7y - 2y²)dy

= 7y²/2 - 2y³/3

= y²(7/2 - 2y/3)

The area is the definite integral

F(7/2) - F(0) = 49/4(7/2 - 7/3) = 14.29