Chapter 13: Q. 35 (page 1015)

Find the volume of the solid bounded above by the given function over the specified region
$f (x, y) = 10 - 2 x + y$ , with $Ω$ the region from Exercise 21

Short Answer

Expert verified

The volume is :

$10 e + \frac{e^{2}}{4} - \frac{49}{4}$ cubic units.

Step by step solution

Step 1. Given information

The function:

$f (x, y) = 10 - 2 x + y$

The region in execise 21

Step 2. To setup integral to find volume

In the given region we can see that

$0 \leq y \leq e^{x} 0 \leq x \leq 1$

So volume of the function over this region is:

$V = \int_{0}^{1} \int_{0}^{e^{x}} f (x, y) d y d x = \int_{0}^{1} \int_{0}^{e^{x}} (10 - 2 x + y) d y d x$

Step 3. To evaluate integral.

First integrate the function with respect to y.

\int_{0}^{e^{x}} (10 - 2 x + y) d y = {[10 y - 2 x y + \frac{y^{2}}{2}]}_{0}^{e^{x}} = (10 e^{x} - 2 x e^{x} + \frac{e^{2 x}}{2}) - 0 = 10 e^{x} - 2 x e^{x} + \frac{e^{2 x}}{2}

Now integrate with respect to x.

$\int_{0}^{1} (10 e^{x} - 2 x e^{x} + \frac{e^{2 x}}{2}) d x = {[10 e^{x} - 2 (x e^{x} - e^{x}) + \frac{e^{2 x}}{4}]}_{0}^{1} = (10 e - 2 (e - e) + \frac{e^{2}}{4}) - (10 - 2 (0 - 1) + \frac{1}{4}) = 10 e - 0 + \frac{e^{2}}{4} - 10 - 2 - \frac{1}{4} = 10 e + \frac{e^{2}}{4} - \frac{49}{4}$

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Find the volume of the solid bounded above by the given function over the specified region
$f (x, y) = 10 - 2 x + y$ , with $Ω$ the region from Exercise 21

Short Answer

Step by step solution

Step 1. Given information

Step 2. To setup integral to find volume

Step 3. To evaluate integral.

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Find the volume of the solid bounded above by the given function over the specified regionf(x,y)=10−2x+y, with Ω the region from Exercise 21

Short Answer

Step by step solution

Step 1. Given information

Step 2. To setup integral to find volume

Step 3. To evaluate integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Decision Maths

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find the volume of the solid bounded above by the given function over the specified region
$f (x, y) = 10 - 2 x + y$ , with $Ω$ the region from Exercise 21