Question

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer${\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{0}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{2}\mathbf{y}}^{\mathbf{4}}\mathit{d}\mathit{x}\mathit{d}\mathit{y}$

Short answer

The required solution is 4.

Step-by-step solution

Defining Integrals

An integral is a mathematical notion that explains displacement, area, volume, and other ideas that come from the combination of infinitesimal data.

 Step 2: Integrating the problem

Integrating with respect to the variable .

$$\begin{gathered} {\int }_{\mathrm{y}=0}^2{\int }_{\mathrm{x}=2\mathrm{y}}^4\mathrm{dxdy}={\int }_{\mathrm{y}=0}^2\left[{\int }_{\mathrm{x}=2\mathrm{y}}^4\mathrm{dx}\right]\mathrm{dy} \\ ={\int }_{\mathrm{y}=0}^2\mathrm{dy}\left(4-2\mathrm{y}\right) \end{gathered}$$

Further Integrating 

Integrating with respect to the variable y.

$$\begin{gathered} {\int }_{y=0}^2\mathrm{dy}\left(4-2\mathrm{y}\right)=4\times \mathrm{y}\overline{){}_0^2-{\mathrm{y}}^2\overline{){}_0^2}} \\ =8\left(2-0\right)-\left(4-0\right) \\ =4 \end{gathered}$$

Therefore, the value is 4.