Q. 69 - Calculus

Question

Let $\mathfrak{R}=\left\{(x,y,z)|a_1\le x\le a_2, b_1\le y\le b_2 ,\text{and} c_1\le z\le c_2\right\}.$If α(x), β( y), and γ (z) are integrable on the intervals $[a_1,a_2],[b_1,b_2], \text{and} [c_1,c_2],$ respectively, use Fubini’s theorem to prove that

${\iiint }_{\mathfrak{R}}\alpha \left(x\right)\beta \left(y\right)\gamma \left(z\right)dV=\left({\int }_{a_1}^{a_2}\alpha \left(x\right)dx\right)\left({\int }_{b_1}^{b_2}\beta \left(y\right)dy\right)\left({\int }_{c_1}^{c_2}\gamma \left(z\right)dz\right).$

Step-by-Step Solution

Answer

The given statement is proved.

Step 1. Given Information.

It is given that $\mathfrak{R}=\left\{(x,y,z)|a_1\le x\le a_2, b_1\le y\le b_2 ,\text{and} c_1\le z\le c_2\right\}.$

Step 2. Prove.

To prove the given statement, we will use Fubini's theorem:

$$\begin{gathered} {\iiint }_{\mathfrak{R}}\alpha \left(x\right)\beta \left(y\right)\gamma \left(z\right)dv={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}\left[{\int }_{c_1}^{c_2}\alpha \left(x\right)\beta \left(y\right)\gamma \left(z\right)dz\right]dydx \\ {\iiint }_{\mathfrak{R}}\alpha \left(x\right)\beta \left(y\right)\gamma \left(z\right)dv={\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}\left[\alpha \left(x\right)\beta \left(y\right){\int }_{c_1}^{c_2}\gamma \left(z\right)dz\right]dydx \end{gathered}$$

Now,

$$\begin{gathered} ={\int }_{a_1}^{a_2}\left[{\int }_{b_1}^{b_2}\alpha \left(x\right)\beta \left(y\right)\left({\int }_{c_1}^{c_2}\gamma \left(z\right)dz\right)dydx\right] \\ =\left({\int }_{a_1}^{a_2}\alpha \left(x\right)dx\right)\left({\int }_{b_1}^{b_2}\beta \left(y\right)dy\right)\left({\int }_{c_1}^{c_2}\gamma \left(z\right)dz)\right) \end{gathered}$$

Hence proved.