evaluate $iint_{b}(x + y),dx,dy$, where $b$ is the rectangle in the $xy$ plane with vertices at…
evaluate $iint_{b}(x + y),dx,dy$, where $b$ is the rectangle in the $xy$ plane with vertices at $(0,1),(1,0),(3,4)$ and $(4,3)$. (enter your answer exactly. use symbolic notation and fractions where needed.) $iint_{b}(x + y),dx,dy=$
Answer
Explanation:
Step1: Find the limits of integration
First, find the equations of the sides of the rectangle to determine the limits. The rectangle has (x) - limits and (y) - limits. By observing the vertices, we can see that (x) ranges from (0) to (4) and (y) ranges from (0) to (4). The double - integral (\iint_{B}(x + y)dxdy=\int_{y_1}^{y_2}\int_{x_1}^{x_2}(x + y)dxdy), where (x_1 = 0,x_2=4,y_1 = 0,y_2 = 4).
Step2: Integrate with respect to (x) first
(\int_{0}^{4}\left[\int_{0}^{4}(x + y)dx\right]dy=\int_{0}^{4}\left[\frac{x^{2}}{2}+yx\right]{x = 0}^{x = 4}dy) [ \begin{align*} &=\int{0}^{4}\left(\frac{4^{2}}{2}+4y-0 - 0\right)dy\ &=\int_{0}^{4}(8 + 4y)dy \end{align*} ]
Step3: Integrate with respect to (y)
(\int_{0}^{4}(8 + 4y)dy=\left[8y+4\times\frac{y^{2}}{2}\right]_{0}^{4}) [ \begin{align*} &=8\times4 + 2\times4^{2}-0-0\ &=32+32\ &=64 \end{align*} ]
Answer:
64