Area between a curve, x axis and a line

Question

I need to calculate the area of the plane between the curve

$y=4\sqrt{x}$, the $x$ axis and the line $y=2x + 2$.

Also, I need to calculate the area using vertical rectangle and another time using horizontal rectangle.

I would like to know how I can start this exercice as I am not sure what is that "rectangle method".

Answer 1

The vertical rectangle means integration with respect to $\mathrm{d}x$, and the horizontal rectangle means integration with respect to $\mathrm{d}y$.

$$A_{\text{vertical rectangle}}=\int^1_0\left(2x+2-4\sqrt{x}\right)\mathrm{d}x+\int^0_{-1}\left(2x+2\right)\mathrm{d}x\\ A_{\text{horizontal rectangle}}=\int^4_0\left(\dfrac{y^2}{16}-\left(\dfrac{y}{2}-1\right)\right)\mathrm{d}y$$