Question

Question: $\mathbf{\int }{\mathbf{\int }}_{\mathbf{A}}\mathit{y}\mathit{d}\mathit{x}\mathit{d}\mathit{y}$where A is the area in Figure 2.8

Short answer

The required solution is$\int {\int }_{\mathrm{A}}\mathrm{ydxdy}=5\mathrm{\pi }$

Step-by-step solution

Defining the Area of integration.

A definite integral between two points can be used to find the area under a curve between two points.

Separating the area of integration.

Divide the integration region into two rectangles and integrate each one separately:

$$\begin{gathered} \int {\int }_{\mathrm{A}}\mathrm{ydxdy}={\int }_0^{\mathrm{\pi }/2}\mathrm{dx}{\int }_0^2\mathrm{ydy}+{\int }_{\mathrm{\pi }/2}^{\mathrm{\pi }}\mathrm{dx}{\int }_0^4\mathrm{ydy} \\ =\frac{\mathrm{\pi }}{4}(4-0)+\frac{\mathrm{\pi }}{4}(16-0) \\ =5\mathrm{\pi } \end{gathered}$$

Therefore, the value is $5\mathrm{\pi }$.