Chapter 13: Q. 44 (page 1015)

Find the volumes of the solids described in Exercises 41–44. The solid bounded above by the hyperboloid with equation $z = x^{2} - y^{2}$ and bounded below by the square with vertices: $(2, 2, - 4), (2, - 2, - 4), (- 2, - 2, - 4), (- 2, 2, - 4) .$

Short Answer

Expert verified

The volume of the described solid is: 0

Step by step solution

Step 1. Given Information

The solid bounded above by the hyperboloid, $z = x^{2} - y^{2}$

The solid bounded below by the square with vertices:

(2, 2, - 4), (2, - 2, - 4), (- 2, - 2, - 4), and (- 2, 2, - 4) .

The region is: $R = {(x, y) | - 2 \leq x \leq 2$ and $- 2 \leq y \leq 2}$

Step 2. Finding the volume of given solid

The double integral to find the volume of the given solid is given by,

\int_{- 2}^{2} \int_{- 2}^{2} (x^{2} - y^{2}) d y d x = \int_{- 2}^{2} {(x^{2} y - \frac{y^{3}}{3})}_{- 2}^{2} d x = \int_{- 2}^{2} (x^{2} (2 - (- 2) - \frac{2^{3} - {(- 2)}^{3}}{3}) d x = \int_{- 2}^{2} (4 x^{2} - \frac{16}{3}) d x = {(\frac{4 x^{3}}{3} - \frac{16 x}{3})}_{- 2}^{2} = \frac{4 (2^{3} - {(- 2)}^{3})}{3} - \frac{16 (2 - (- 2))}{3} = \frac{64}{3} - \frac{64}{3} = 0

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Find the volumes of the solids described in Exercises 41–44. The solid bounded above by the hyperboloid with equation $z = x^{2} - y^{2}$ and bounded below by the square with vertices: $(2, 2, - 4), (2, - 2, - 4), (- 2, - 2, - 4), (- 2, 2, - 4) .$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the volume of given solid

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Find the volumes of the solids described in Exercises 41–44. The solid bounded above by the hyperboloid with equationz=x2−y2and bounded below by the square with vertices:(2,2,−4),(2,−2,−4),(−2,−2,−4),(−2,2,−4).

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the volume of given solid

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Find the volumes of the solids described in Exercises 41–44. The solid bounded above by the hyperboloid with equation $z = x^{2} - y^{2}$ and bounded below by the square with vertices: $(2, 2, - 4), (2, - 2, - 4), (- 2, - 2, - 4), (- 2, 2, - 4) .$