Question

${\mathbf{\int }}_{\mathbf{y}\mathbf{=}\mathbf{2}}^{\mathbf{3}}{\mathbf{\int }}_{\mathbf{z}\mathbf{=}\mathbf{1}}^{\mathbf{2}}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{y}\mathbf{+}\mathbf{z}}^{\mathbf{2}\mathbf{y}\mathbf{+}\mathbf{z}}\mathbf{6}\mathbf{ydxdzdy}$

Short answer

The solution for the given integral is 70.

Step-by-step solution

Definition of integration

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

Given Integral

The integral${\int }_{\mathrm{y}=2}^3{\int }_{\mathrm{z}=1}^2{\int }_{\mathrm{x}=\mathrm{y}+\mathrm{z}}^{2\mathrm{y}+\mathrm{z}}6\mathrm{ydxdzdy}$.

To find the value for the given integral.

Finding the integral value

Integrate the given value,

$$\begin{gathered} \mathrm{l}={\int }_{-2}^{3}6\mathrm{ydy}{\int }_1^2\mathrm{dz}{\int }_{\mathrm{y}+\mathrm{z}}^{2\mathrm{y}+2}\mathrm{dx} \\ ={\int }_{-2}^{3}6\mathrm{ydy}{\int }_1^2\mathrm{dz}\left(2\mathrm{y}+\mathrm{z}-\mathrm{y}-\mathrm{z}\right) \end{gathered}$$

Further Integrating

Now,

$$\begin{gathered} \mathrm{l}={\int }_{-2}^{3}6\mathrm{ydy}{\int }_1^{2}y\mathrm{dz} \\ ={\int }_{-2}^{3}6y^{2}dy\times 1 \\ ={\overline{)2y^3}}_{-2}^3 \\ =70 \end{gathered}$$

Therefore, the required solution for the given integral is 70.