Q. 2 - Calculus

Question

Using the definition to evaluate a double integral: Evaluate the given double integrals as a limit of a Riemann sum. For each integral, let

$R=\left\{(x,y)|0\le x\le 2 and 1\le y\le 4\right\}$.

$\iint \left(x{y}^2\right)dA$

Step-by-Step Solution

Answer

$\iint \left(x{y}^2\right)dA=42$ square units

Step 1: Definition of double integral

Let $R=\left\{(x,y)|a\le x\le b and c\le y\le d\right\}$ and $f(x,y)$ is continuous on R. Then.

$\iint f(x,y)dA={\int }_a^b{\int }_c^{d}f(x,y)dxdy$

Step 2: Evaluate the integral

$$\begin{gathered} \iint \left(x{y}^2\right)dA={\int }_1^4{\int }_0^2\left(x{y}^2\right)dxdy \\ ={\int }_1^4\left[{\int }_0^2\left(x{y}^2\right)dx\right]dy \\ ={\int }_1^{4}dy{\left[\frac{x^2}{2}{y}^2\right]}_0^2 \\ ={\int }_1^{4}dy\left[\frac{2^2}{2}{y}^2\right] \\ ={\int }_1^4\left(2y^2\right)dy \\ ={\left[2\frac{y^3}{3}\right]}_1^4 \\ =\frac{128}{3}-\frac{2}{3} \\ =\frac{126}{3} \\ =42 \end{gathered}$$