Explanation Given information: The solid bounded by the region x = 2 − y 2 and x = y 4 . The region is about the y -axis. The region lies between a = − 1 and b = 1 . Calculation: Show the Equations of the solid region. x = y 4 (1) x = 2 − y 2 (2) Substitute − 1 for y in Equation (1). x = ( − 1 ) 4 = 1 Substitute 0 for y in Equation (1). x = ( 0 ) 4 = 0 Substitute 1 for y in Equation (1). x = ( 1 ) 4 = 1 Substitute − 1 for y in Equation (2). x = 2 − ( − 1 ) 2 = 1 Substitute 0 for y in Equation (2). x = 2 − ( 0 ) 2 = 2 Substitute 1 for y in Equation (2). x = 2 − ( 1 ) 2 = 1 Sketch the region for the equations x = y 4 and x = 2 − y 2 using the calculated values as shown in Figure 1. Refer to Figure 1. Consider that the solid is a washer which is obtained by rotating region. Consider that the cross section is a washer with outer radius 2 − y 2 and inner radius y 4 . Sketch the cross section of the washer as shown in Figure 2. Find the area of the washer as shown below. A ( y ) = π r 1 2 − π r 2 2 = π ( 2 − y 2 ) 2 − π × ( y 4 ) 2 = π ( 4 + y 4 − 4 y 2 − y 8 ) The expression to find the volume of the washer as shown below

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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 5.2, Problem 10E

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 5.2, Problem 9E

Chapter 5.2, Problem 11E

Students have asked these similar questions

nd ave a ction and ave an 48. The domain of f y=f'(x) x 1 2 (= x<0 x<0 = f(x) possible. Group Activity In Exercises 49 and 50, do the following. (a) Find the absolute extrema of f and where they occur. (b) Find any points of inflection. (c) Sketch a possible graph of f. 49. f is continuous on [0,3] and satisfies the following. X 0 1 2 3 f 0 2 0 -2 f' 3 0 does not exist -3 f" 0 -1 does not exist 0 ve tes where X 0 < x <1 1< x <2 2

Numerically estimate the value of limx→2+x3−83x−9, rounded correctly to one decimal place. In the provided table below, you must enter your answers rounded exactly to the correct number of decimals, based on the Numerical Conventions for MATH1044 (see lecture notes 1.3 Actions page 3). If there are more rows provided in the table than you need, enter NA for those output values in the table that should not be used. x→2+ x3−83x−9 2.1 2.01 2.001 2.0001 2.00001 2.000001

Find the general solution of the given differential equation. (1+x)dy/dx - xy = x +x2

Chapter 5 Solutions

Single Variable Calculus

Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 6E Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 8E Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....

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 calculates the volume of the solid obtained by rotating region using the calculated values as shown in Figure 1.

Concept explainers

Videos

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

Want to see the full answer?

Chapter 5 Solutions