Question

${\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{0}}^{\mathbf{1}}{\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{2}}^{\mathbf{4}}\mathbf{3}\mathbf{xdydx}$

Short answer

The solution for the given integral${\int }_{\mathrm{x}=0}^1{\int }_{\mathrm{y}=2}^{4}3\mathrm{xdydx}$ is 3.

Step-by-step solution

Definition of integration

Integration is a way of finding the whole by adding or summing the parts.

Given quantities

The integral${\int }_{\mathrm{x}=0}^1{\int }_{\mathrm{y}=2}^{4}3\mathrm{x}\mathrm{dydx}$,

To find the value for the given integral.

Finding the integral value

Integrate the given value,

$$\begin{gathered} {\int }_{\mathrm{x}=0}^1{\int }_{\mathrm{y}=2}^{4}3\mathrm{x}\mathrm{dydx}=\left(\mathrm{y}\overline{){}_2^4}\right)\left(\frac{3}{2}{\mathrm{x}}^2\overline{){}_0^1}\right) \\ =2\times \frac{3}{2} \\ =3 \end{gathered}$$

Therefore, the required solution for the given integral is ${\int }_{\mathrm{x}=0}^1{\int }_{\mathrm{y}=2}^{4}3\mathrm{x}\mathrm{dydx}$is 3.