Chapter 13: Q. 62 (page 1005)

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = \frac{y}{x} e^{y} W j e r e R = {(x, y) | 1 \leq x \leq e, 0 \leq y \leq 2}$

Short Answer

Expert verified

Volume of the integral is $e^{2} - 1 c u b i c u n i t s$

Step by step solution

Given information

We are given a region

$f (x, y) = \frac{y}{x} e^{y} W j e r e R = {(x, y) | 1 \leq x \leq e, 0 \leq y \leq 2}$

Evaluate the integral

We have

$\int_{R} \frac{y}{x} e^{y} d A = \int_{0}^{2} \int_{1}^{e} \frac{y}{x} e^{y} d x d y = \int_{0}^{2} {[\ln x]}^{e}_{1} \cdot y e^{y} d y = \int_{0}^{2} y e^{y} d y = {[y e^{y} - e^{y}]}^{2}_{0} = e^{2} - 1$

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Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = \frac{y}{x} e^{y} W j e r e R = {(x, y) | 1 \leq x \leq e, 0 \leq y \leq 2}$

Short Answer

Step by step solution

Given information

Evaluate the integral

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Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane f(x,y)=yxeyWjereR={(x,y)|1≤x≤e,0≤y≤2}

Short Answer

Step by step solution

Given information

Evaluate the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Decision Maths

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Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = \frac{y}{x} e^{y} W j e r e R = {(x, y) | 1 \leq x \leq e, 0 \leq y \leq 2}$