Chapter 13: Q. 36 (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 22.

Short Answer

Expert verified

The volume of the function is: $54$ cubic inches.

Step by step solution

Step 1. Given information

The function:

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

The region in exercise 22

Step 2. To setup integral

In the given region the $- x \leq y \leq x$ and $0 \leq x \leq 3$

So, the volume of the function over specified region is:

$V = \int_{0}^{3} \int_{- x}^{x} f (x, y) d y d x = \int_{0}^{3} \int_{- x}^{x} (10 - 2 x + y) d y d x$

Step 3. Evaluate integral

First integrate the function with respect to y.

\int_{- x}^{x} (10 - 2 x + y) d y = {[10 y - 2 x y + \frac{y^{2}}{2}]}_{- x}^{x} = (10 x - 2 x^{2} + \frac{x^{2}}{2}) - (- 10 x + 2 x^{2} + \frac{x^{2}}{2}) = 20 x - 4 x^{2}

Now integrate the above function of x with respect to x.

$\int_{0}^{3} (20 x - 4 x^{2}) d x = {[20 \frac{x^{2}}{2} - 4 \frac{x^{3}}{3}]}_{0}^{3} = {[10 x^{2} - \frac{4}{3} x^{3}]}_{0}^{3} = [10 {(3)}^{2} - \frac{4}{3} {(3)}^{3} - 0] = 90 - 36 = 54$

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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 22.

Short Answer

Step by step solution

Step 1. Given information

Step 2. To setup integral

Step 3. Evaluate integral

One App. One Place for Learning.

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find the volume of the solid bounded above by the given function over the specified region f(x,y)=10−2x+y, withΩthe region from Exercise 22.

Short Answer

Step by step solution

Step 1. Given information

Step 2. To setup integral

Step 3. Evaluate integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Probability and Statistics

Calculus

Applied Mathematics

Discrete Mathematics

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 22.