Question

Find the area bounded by the curves y = x3, y = 1, and x = 0 using double integration.

Answer 1

From the graph: $0<x<y^{\frac{1}{3}}, 0<y<1$ [Since $y=x^3\Rightarrow x=y^{\frac{1}{3}}$ ]

Area

$=\int_{0}^{1}\int_{0}^{y^{\frac 1 3}} dxdy=\int_{0}^{1}[x]_{0}^{y^{\frac{1}{3}}}dy=\int_{0}^{1} y^{\frac{1}{3}}dy\\ =[\frac{y^{\frac 4 3}}{\frac{4}{3}}]_{0}^{1}=\frac 3 4=0.75\ \text{sq. units}$