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 $R=\left\{(x,y)|0\le x\le 2and1\le y\le 4\right\}$.

$\iint (x+2y)dA$

Short answer

$\iint (x+2y)dA=36squareunits$

Step-by-step solution

Definition of double integral

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

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

Evaluate the integral

$$\begin{gathered} \iint (x+2y)dA={\int }_1^4{\int }_0^2(x+2y)dxdy \\ ={\int }_1^4\left[{\int }_0^2(x+2y)dx\right]dy \\ ={\int }_1^{4}dy{\left[\frac{x^2}{2}+2xy\right]}_0^2 \\ ={\int }_1^{4}dy\left[\frac{2^2}{2}+2\times 2\times y\right] \\ ={\int }_1^4(2+4y)dy \\ ={\left[2y+4\frac{y^2}{2}\right]}_1^4 \\ =\left(8+32\right)-\left(2+2\right) \\ =40-4 \\ =36 \end{gathered}$$