Explanation Given information: The solid bounded by the region x 2 − y 2 = a 2 and x = a + h about the y axis is ( a > 0 , h > 0 ) . Find the intersection points of the curves as shown below: x 2 − y 2 = a 2 y 2 = x 2 − a 2 y = ± x 2 − a 2 Consider the condition x = a ( a > 0 ) . The region enclosed by the curves y = x 2 − a 2 and y = − x 2 − a 2 shown in Figure 1. Refer to Figure (1). Consider that the solid is a washer which is obtained by rotating region, about the line y axis. Calculate the volume rotate about y –axis. V = 2 π ∫ a a + h x y 2 d x − 2 π ∫ a a + h x y 1 d x (1) Here, a is lower limit for the integral and a + h is upper limit for the integral Substitute x 2 − a 2 for y 2 , and − x 2 − a 2 for y 2 in Equation (1) V = 2 π ∫ a a + h x x 2 − a 2 d x − 2 π ∫ a a + h x ( − x 2 − a 2 ) d x Modify Equation (1), V = 2 π ∫ a a + h x x 2 − a 2 d x + 2 π ∫ a a + h x ( x 2 − a 2 ) d x = 4 π ∫ a a + h x x 2 − a 2 d x Consider u = x 2 − a 2 (2) Differentiate Equation (2) with respect to x

9th Edition

ISBN: 9781337613927

Author: Stewart

Publisher: CENGAGE L

See similar textbooks

Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Videos

Question

Chapter 6, Problem 11E

To determine

To calculate: The volume of the solid obtained by rotating region.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6, Problem 10E

Chapter 6, Problem 12E

Students have asked these similar questions

If a snowball melts so that its surface area decreases at a rate of 10 cm²/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 12 cm. (Round your answer to three decimal places.) cm/min

1) let X: N R be a sequence and let Y: N+R be the squence obtained from x by di scarding the first meN terms of x in other words Y(n) = x(m+h) then X converges to L If and only is y converges to L- 11) let Xn = cos(n) where nyo prove D2-1 that lim xn = 0 by def. h→00 ii) prove that for any irrational numbers ther exsist asquence of rational numbers (xn) converg to S.

4.2 Product and Quotient Rules 1. 9(x)=125+1 y14+2 Use the product and/or quotient rule to find the derivative of each function. a. g(x)= b. y (2x-3)(x-1) c. y== 3x-4 √x

Chapter 6 Solutions

Calculus, Early Transcendentals

Ch. 6.1 - (a) Set up an integral for the area of the shaded...Ch. 6.1 - (a) Set up an integral for the area of the shaded...Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region. 5.Ch. 6.1 - Find the area of the shaded region. 6.Ch. 6.1 - Set up, but do not evaluate, an integral...Ch. 6.1 - Prob. 8E Ch. 6.1 - Set up, but do not evaluate, an integral...Ch. 6.1 - Set up, but do not evaluate, an integral...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The volume of the solid obtained by rotating region.

Solution Summary: The author explains how to calculate the volume of the solid obtained by rotating region.

Concept explainers

Videos

To calculate: The volume of the solid obtained by rotating region.

Want to see the full answer?

Chapter 6 Solutions