Q 19. - Calculus

Question

Express the area of the region  between the function $f\left(x\right)=x^3$ and the $x$-axis on the interval $[-3, 3]$ as a sum of two iterated integrals, integrating first with respect to $x$ in each. Express the area of  as a sum of two different iterated integrals, integrating first with respect to y. Now evaluate your integrals.

Step-by-Step Solution

Answer

The area of region is zero.

Step 1: Given Information

It is given that and interval is $\left[-3,3\right]$.

Step 2: Region of Integration

The region of integration is given as

Step 3: Limits

First region of integration ${\Omega }_1$ is bounded by $x=y^{1/3}, x=-3, x$ axis

$\Rightarrow -3\le x\le y^{1/3}, -27\le y\le 0$

Second region ${\Omega }_2$ is bounded by curve $x=y^{1/3}$ and line $x=3$ and $x$ axis

$\Rightarrow y^{1/3}\le x\le 3$ and $0\le y\le 27$