Chapter 13: Q. 69 (page 1057)

Let $R = \{(x, y, z) | a_{1} 鈮� x 鈮� a_{2}, b_{1} 鈮� y 鈮� b_{2}, and c_{1} 鈮� z 鈮� c_{2}\} .$ If 伪(x), 尾( y), and 纬 (z) are integrable on the intervals $[a_{1}, a_{2}], [b_{1}, b_{2}], and [c_{1}, c_{2}],$ respectively, use Fubini鈥檚 theorem to prove that
${鈭�}_{R} 伪 (x) 尾 (y) 纬 (z) d V = ({鈭�}_{a_{1}}^{a_{2}} 伪 (x) d x) ({鈭�}_{b_{1}}^{b_{2}} 尾 (y) d y) ({鈭�}_{c_{1}}^{c_{2}} 纬 (z) d z) .$

Short Answer

Expert verified

The given statement is proved.

Step by step solution

Step 1. Given Information.

It is given that $R = \{(x, y, z) | a_{1} 鈮� x 鈮� a_{2}, b_{1} 鈮� y 鈮� b_{2}, and c_{1} 鈮� z 鈮� c_{2}\} .$

Step 2. Prove.

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

${鈭�}_{R} 伪 (x) 尾 (y) 纬 (z) d v = {鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} [{鈭�}_{c_{1}}^{c_{2}} 伪 (x) 尾 (y) 纬 (z) d z] d y d x {鈭�}_{R} 伪 (x) 尾 (y) 纬 (z) d v = {鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} [伪 (x) 尾 (y) {鈭�}_{c_{1}}^{c_{2}} 纬 (z) d z] d y d x$

Now,

role="math" localid="1650361075356" $= {鈭�}_{a_{1}}^{a_{2}} [{鈭�}_{b_{1}}^{b_{2}} 伪 (x) 尾 (y) ({鈭�}_{c_{1}}^{c_{2}} 纬 (z) d z) d y d x] = ({鈭�}_{a_{1}}^{a_{2}} 伪 (x) d x) ({鈭�}_{b_{1}}^{b_{2}} 尾 (y) d y) ({鈭�}_{c_{1}}^{c_{2}} 纬 (z) d z))$

Hence proved.

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Prove.

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Let R=(x,y,z)|a1鈮�x鈮�a2,b1鈮�y鈮�b2,andc1鈮�z鈮�c2.If 伪(x), 尾( y), and 纬 (z) are integrable on the intervals [a1,a2],[b1,b2],and[c1,c2],respectively, use Fubini鈥檚 theorem to prove that鈭�R伪(x)尾(y)纬(z)dV=鈭�a1a2伪(x)dx鈭�b1b2尾(y)dy鈭�c1c2纬(z)dz.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Prove.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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