202109161150 Exercises 3.3.B (Q9)

Let $M$ be a constant. Show that

$\displaystyle{\int_{z_1}^{z_2}\int_{y_1}^{y_2}\int_{x_1}^{x_2}M\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z}=M(z_2-z_1)(y_2-y_1)(x_2-x_1)$ .

Solution.

$\begin{aligned} & \qquad \int_{z_1}^{z_2}\int_{y_1}^{y_2}\int_{x_1}^{x_2}M\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \\ & = M\int_{z_1}^{z_2}\int_{y_1}^{y_2}\big[ x\big]\big|_{x_1}^{x_2} \,\mathrm{d}y\,\mathrm{d}z \\ & = M(x_2-x_1)\int_{z_1}^{z_2}\int_{y_1}^{y_2} \,\mathrm{d}y\,\mathrm{d}z \\ & = M(x_2-x_1)\int_{z_1}^{z_2}\big[ y\big] \big|_{y_1}^{y_2}\,\mathrm{d}z \\ & = M(y_2-y_1)(x_2-x_1)\int_{z_1}^{z_2}\,\mathrm{d}z \\ & = M(y_2-y_1)(x_2-x_1)\big[ z\big] \big|_{z_1}^{z_2} \\ & = M(z_2-z_1)(y_2-y_1)(x_2-x_1) \\ \end{aligned}$

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20	21	22	23	24	25	26
27	28	29	30

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