Example 2 | Plane Areas in Rectangular Coordinates

$a^2 \, y = x^3$

when $x = 0, \,\, y = 0$

when $x = 2a, \,\, y = 8a$

points of intersection: (0, 0) and (2a, 8a)
 

Using vertical strip:
$\displaystyle A = {\int_{x_1}}^{x_2} y \, dx$

$\displaystyle A = {\int_0}^{2a} \dfrac{x^3 \, dx}{a^2}$

$A = \left[ \dfrac{x^4}{4a^2} \right]_0^{2a}$

$A = \dfrac{(2a)^4 - 0^4}{4a^2}$

$A = 4a^2 \,\, \text{unit}^2$           answer