Math

CalculusSingle Variable Calculus: Early Transcendentals(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x -axis and (ii) the y -axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. 5. x = y + y 3 , 0 ≤ y ≤ 1BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305270336

Chapter 8.2, Problem 5E

Textbook Problem

- (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the
*x*-axis and (ii) the*y*-axis. - (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.

**5.** *x* = *y* + *y*^{3}, 0 ≤ *y* ≤ 1

Expert Solution

(a) (i)

To determine

**To set up:** An integral for the area of the surface obtained by rotating the curve about *x*-axis.

**Given information:**

The equation of the curve is

**Calculation:**

Show the equation of the curve.

Calculate the area of the surface obtained by rotating the curve about *x*-axis:

Here, *S* is the area of the surface obtained by rotating the curve about *x*-axis and

Differentiate both sides of Equation (1) with respect to *y*

Expert Solution

(ii)

To determine

**To set up**: An integral for the area of the surface obtained by rotating the curve about *y*-axis.

Expert Solution

(b) (i)

To determine

**To evaluate:** the surface area of the curve about *x*-axis by using numerical integration capability of the calculator.

Expert Solution

(ii)

To determine

**To evaluate:** the surface area of the curve about *y*-axis by using numerical integration capability of the calculator.

Single Variable Calculus: Early Transcendentals

Show all chapter solutions

Ch. 8.1 - Use the arc length formula (3) to find the length...Ch. 8.1 - Use the arc length formula to find the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Find the exact length of the curve. 9. y = 1 +...Ch. 8.1 - Find the exact length of the curve. 10. 36y2 = (x2...

Ch. 8.1 - Find the exact length of the curve. 11....Ch. 8.1 - Find the exact length of the curve. 12....Ch. 8.1 - Find the exact length of the curve. 13....Ch. 8.1 - Find the exact length of the curve. 14. y = ln(cos...Ch. 8.1 - Find the exact length of the curve. 15. y = ln(sec...Ch. 8.1 - Find the exact length of the curve. 16....Ch. 8.1 - Find the exact length of the curve. 17....Ch. 8.1 - Find the exact length of the curve. 18....Ch. 8.1 - Find the exact length of the curve. 19....Ch. 8.1 - Find the exact length of the curve. 20. y = 1 ex,...Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - (a) Graph the curve y=x4x3,0x4. (b) Compute the...Ch. 8.1 - Repeat Exercise 29 for the curve y=x+sinx0x2Ch. 8.1 - Sketch the curve with equation x2/3 + y2/3 = 1 and...Ch. 8.1 - (a) Sketch the curve y3 = x2. (b) Use Formulas 3...Ch. 8.1 - Find the arc length function for the curve y =...Ch. 8.1 - (a) Find the arc length function for the curve y =...Ch. 8.1 - Find the arc length function for the curve...Ch. 8.1 - The arc length function for a curve y = f(x),...Ch. 8.1 - For the function f(x)=14ex+ex, prove that the arc...Ch. 8.1 - A steady wind blows a kite due west. The kites...Ch. 8.1 - A hawk flying at 15 m/s at an altitude of 180 m...Ch. 8.1 - The Gateway Arch in St. Louis (see the photo on...Ch. 8.1 - A manufacturer of corrugated metal roofing wants...Ch. 8.1 - (a) The figure shows a telephone wire hanging...Ch. 8.1 - Find the length of the curve y=1xt31dt1x4Ch. 8.1 - The curves with equations x + y = l , n = 4, 6, 8,...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use either a CAS or a table of integrals to find...Ch. 8.2 - Use either a CAS or a table of integrals to find...Ch. 8.2 - If the region R=(x,y)x1,0y1/x is rotated about the...Ch. 8.2 - If the infinite curve y = ex, x 0, is rotated...Ch. 8.2 - (a) If a 0, find the area of the surface...Ch. 8.2 - A group of engineers is building a parabolic...Ch. 8.2 - (a) The ellipsex2a2+y2b2=1abis rotated about the...Ch. 8.2 - Find the surface area of the torus in Exercise...Ch. 8.2 - If the curve y = f(x), a x b, is rotated about...Ch. 8.2 - Find the area of the surface obtained by rotating...Ch. 8.2 - (a) Show that the surface area of a zone of a...Ch. 8.2 - Show that if we rotate the curve y = ex/2 + ex/2...Ch. 8.2 - Let L be the length of the curve y = f(x), a x ...Ch. 8.2 - Formula 4 is valid only when f(x) 0. Show that...Ch. 8.3 - An aquarium 5 ft long, 2 ft wide, and 3 ft deep is...Ch. 8.3 - A tank is 8 m long, 4 m wide, 2 m high, and...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A milk truck carries milk with density 64.6 lb/ft3...Ch. 8.3 - A trough is filled with a liquid of density 840...Ch. 8.3 - A vertical dam has a semicircular gate as shown in...Ch. 8.3 - A cube with 20-cm-long sides is sitting on the...Ch. 8.3 - A dam is inclined at an angle of 30 from the...Ch. 8.3 - A swimming pool is 20 ft wide and 40 ft long and...Ch. 8.3 - Suppose that a plate is immersed vertically in a...Ch. 8.3 - A metal plate was found submerged vertically in...Ch. 8.3 - (a) Use the formula of Exercise 18 to show that...Ch. 8.3 - Point-masses mi are located on the x-axis as...Ch. 8.3 - Point-masses mi are located on the x-axis as...Ch. 8.3 - The masses mi are located at the points Pi. Find...Ch. 8.3 - The masses mi are located at the points Pi. Find...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Use Simpsons Rule to estimate the centroid of the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Use a graph to find approximate x-coordinates of...Ch. 8.3 - Prove that the centroid of any triangle is located...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - A rectangle with sides a and b is divided into two...Ch. 8.3 - If x is the x-coordinate of the centroid of the...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - The centroid of a curve can be found by a process...Ch. 8.3 - The Second Theorem of Pappus is in the same spirit...Ch. 8.3 - Use the Second Theorem of Pappus described in...Ch. 8.3 - Let R be the region that lies between the curves...Ch. 8.3 - Prove Formulas 9.Ch. 8.4 - The marginal cost function C(x) was defined to be...Ch. 8.4 - A company estimates that the marginal revenue (in...Ch. 8.4 - A mining company estimates that the marginal cost...Ch. 8.4 - The demand function for a particular vacation...Ch. 8.4 - A demand curve is given by p = 450/(x + 8). Find...Ch. 8.4 - The supply function ps(x) for a commodity gives...Ch. 8.4 - If a supply curve is modeled by the equation p =...Ch. 8.4 - In a purely competitive market, the price of a...Ch. 8.4 - The sum of consumer surplus and producer surplus...Ch. 8.4 - A camera company estimates that the demand...Ch. 8.4 - A company modeled the demand curve for its product...Ch. 8.4 - A movie theater has been charging 10.00 per person...Ch. 8.4 - If the amount of capital that a company has at...Ch. 8.4 - If revenue flows into a company at a rate of...Ch. 8.4 - If income is continuously collected at a rate of...Ch. 8.4 - The present value of an income stream is the...Ch. 8.4 - Paretos Law of Income states that the number of...Ch. 8.4 - A hot, wet summer is causing a mosquito population...Ch. 8.4 - Use Poiseuilles Law to calculate the rate of flow...Ch. 8.4 - High blood pressure results from constriction of...Ch. 8.4 - The dye dilution method is used to measure cardiac...Ch. 8.4 - After a 5.5-mg injection of dye, the readings of...Ch. 8.4 - The graph of the concentration function c(t) is...Ch. 8.5 - Let f (x) be the probability density function for...Ch. 8.5 - Let f(t) be the probability density function for...Ch. 8.5 - Let f (x) = 30x2(1 x)2 for 0 x 1 and f (x) = 0...Ch. 8.5 - The density function f(x)=e3x(1+e3x)2 is an...Ch. 8.5 - Let f (x) = c/(1 + x2). (a) For what value of c is...Ch. 8.5 - Let f(x) = k (3x x2) if 0 x 3 and f(x) = 0 if x...Ch. 8.5 - A spinner from a board game randomly indicates a...Ch. 8.5 - (a) Explain why the function whose graph is shown...Ch. 8.5 - Show that the median waiting time for a phone call...Ch. 8.5 - (a) A type of light bulb is labeled as having an...Ch. 8.5 - An online retailer has determined that the average...Ch. 8.5 - The time between infection and the display of...Ch. 8.5 - REM sleep is the phase of sleep when most active...Ch. 8.5 - According to the National Health Survey, the...Ch. 8.5 - The Garbage Project at the University of Arizona...Ch. 8.5 - Boxes are labeled as containing 500 g of cereal....Ch. 8.5 - The speeds of vehicles on a highway with speed...Ch. 8.5 - Show that the probability density function for a...Ch. 8.5 - For any normal distribution, find the probability...Ch. 8.5 - The standard deviation for a random variable with...Ch. 8.5 - The hydrogen atom is composed of one proton in the...Ch. 8 - (a) How is the length of a curve defined? (b)...Ch. 8 - (a) Write an expression for the surface area of...Ch. 8 - Describe how we can find the hydrostatic force...Ch. 8 - (a) What is the physical significance of the...Ch. 8 - What does the Theorem of Pappus say?Ch. 8 - Given a demand function p(x), explain what is...Ch. 8 - (a) What is the cardiac output of the heart? (b)...Ch. 8 - What is a probability density function? What...Ch. 8 - Suppose f(x) is the probability density function...Ch. 8 - What is a normal distribution? What is the...Ch. 8 - Find the length of the curve. 1. y = 4 (x 1)3/2,...Ch. 8 - Find the length of the curve. 2.y=2ln(sin12x), /3 ...Ch. 8 - Find the length of the curve. 3. 12x = 4y3 + 3y1,...Ch. 8 - (a) Find the length of the curve y=x416+12x21x2...Ch. 8 - Let C be the arc of the curve y = 2/(x + 1) from...Ch. 8 - (a) The curve y = x2, 0 x 1, is rotated about...Ch. 8 - Use Simpsons Rule with n = 10 to estimate the...Ch. 8 - Use Simpsons Rule with n = 10 to estimate the area...Ch. 8 - Find the length of the curve y=1xt1dt1x16Ch. 8 - Find the area of the surface obtained by rotating...Ch. 8 - A gate in an irrigation canal is constructed in...Ch. 8 - A trough is filled with water and its vertical...Ch. 8 - Find the centroid of the region shown. 13.Ch. 8 - Find the centroid of the region shown. 14.Ch. 8 - Find the centroid of the region bounded by the...Ch. 8 - Find the centroid of the region bounded by the...Ch. 8 - Find the volume obtained when the circle of radius...Ch. 8 - Use the Theorem of Pappus and the fact that the...Ch. 8 - The demand function for a commodity is given by...Ch. 8 - After a 6-mg injection of dye into a heart, the...Ch. 8 - (a) Explain why the function...Ch. 8 - Lengths of human pregnancies are normally...Ch. 8 - The length of time spent waiting in line at a...Ch. 8 - Find the area of the region S = {(x, y) | x 0, y ...Ch. 8 - Find the centroid of the region enclosed by the...Ch. 8 - If a sphere of radius r is sliced by a plane whose...Ch. 8 - (a) Show that an observer at height H above the...Ch. 8 - Suppose that the density of seawater, = (z),...Ch. 8 - The figure shows a semicircle with radius 1,...Ch. 8 - Let P be a pyramid with a square base of side 2b...Ch. 8 - Consider a flat metal plate to be placed...Ch. 8 - A uniform disk with radius 1 m is to be cut by a...Ch. 8 - A triangle with area 30 cm2 is cut from a corner...Ch. 8 - In a famous 18th-century problem, known as Buffons...Ch. 8 - If the needle in Problem 11 has length h L, its...Ch. 8 - Find the centroid of the region enclosed by the...

Find more solutions based on key concepts

Show solutions Finding the Area of a Polar Region In Exercises 97-100, find the area of the region. One petal of r=2sin6

Calculus: Early Transcendental Functions

In Exercises 4756, solve the given equation for the indicated variable. 10,000=700(1.043t+1) (Round the answer ...

Finite Mathematics and Applied Calculus (MindTap Course List)

Show that the curve r = 2 csc (also a conchoid) has the line y = 1 as a horizontal asymptote by showing that...

Calculus: Early Transcendentals

Solve each equation in Exercises 107120 for x, rounding your answer to four significant digits where necessary....

Applied Calculus

Fill the blanks. 10. a. If f(a) is defined limxa f(x) exists and limxaf(x) = f(a), then f is __________ at a. b...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Graphing Functions Sketch a graph of the function by first making a table of values. 18. k(x)=x3

Precalculus: Mathematics for Calculus (Standalone Book)

Sketching the Graph of an Equation In Exercises 19-28, sketch the graph of the equation. Use a graphing utility...

Calculus: An Applied Approach (MindTap Course List)

If u and v are differentiable vector functions, c is a scalar, and f is a real-valued function, write the rules...

Multivariable Calculus

Prove the identity. 11. sinh(x + y) = sinh x cosh y + cosh x sinh y

Single Variable Calculus: Early Transcendentals, Volume I

Rewrite the integral 11x2101yf(x,y,z)dzdydx as an iterated integral in the order dxdydz.

Calculus (MindTap Course List)

The toll T charged for driving on a certain stretch of a toll road is 5 except during rush hours (between 7 am ...

Single Variable Calculus

For Problems 43-50, replace each question mark to make the given statement an application of the indicated prop...

Intermediate Algebra

In Problem 1-10, find the real solutions to each quadratic equation.

Mathematical Applications for the Management, Life, and Social Sciences

A group of quiz, scares, is shown in a histogram If the bars in the histogram gradually decrease in height from...

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Write each complex number in trigonometric form, once using degrees and once using radians. In each case, begin...

Trigonometry (MindTap Course List)

Calculate the annual interest and current yield of the following bonds.
Company Coupon Rate Annual Interest Mar...

Contemporary Mathematics for Business & Consumers

A population has a mean of =50 and a standard deviation of =20 . Would a score of X=70 be considered an extreme...

Statistics for The Behavioral Sciences (MindTap Course List)

PA The city manager has received a complaint from the local union of firefighters to the effect that they are u...

Essentials Of Statistics

The circumference of a circle is 66 ft. aFind the diameter of the circle using 227. bFind the area of the circl...

Elementary Geometry For College Students, 7e

Find the unknown sides of each right triangle (see Illustration 1): ILLUSTRATION 1 c=289.5cm,A=24.63

Elementary Technical Mathematics

Use inductive reasoning to decide whether each statement is correct. Note: The numbers 1, 2, 3, 4, 5, are calle...

Mathematical Excursions (MindTap Course List)

If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type he...

Elements Of Modern Algebra

Expand Your Knowledge: Confidence Intervals for d Using techniques from Section 8.2, we can find a confidence i...

Understanding Basic Statistics

The article Monte Carlo SimulationTool for Better Understanding of LRFD (J. of Structural Engr., 1993: 15861599...

Probability and Statistics for Engineering and the Sciences

Evaluate the expression sin Exercises 116. (14)2

Finite Mathematics

Volume of a Torus In Exercises 55 and 56, find the volume of the torus generated by revolving the region bounde...

Calculus: Early Transcendental Functions (MindTap Course List)

Coterminal Angles in Radians In Exercises 7 and 8, determine two coterminal angles in radian measure (one posit...

Calculus of a Single Variable

In exercises 1316, refer to the following experiment: Two cards are drawn in succession without replacement fro...

Finite Mathematics for the Managerial, Life, and Social Sciences

In TUV not shown, TUTVVU. Write a three-part inequality that compares the measures of the three angles of TUV. ...

Elementary Geometry for College Students

The curvature of at t = 0 is:

Study Guide for Stewart's Multivariable Calculus, 8th

Cylindrical-to-Rectangular ConversionIn Exercises 2330, find an equation in rectangular coordinates for the sur...

Multivariable Calculus

True or False:
If f(x) is continuous and decreasing, f(n) = an for all n = 1, 2, 3, …, and

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Data from the Office for National Statistics show that the mean age at which men in Great Britain get married i...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

In addition to the key words, you should also be able to define each of the following terms: target population ...

Research Methods for the Behavioral Sciences (MindTap Course List)

Consider the following hypothesis test: H0: 80 Ha: 80 A sample of 100 is used and the population standard d...

Statistics for Business & Economics, Revised (MindTap Course List)

It is common for baseball pitchers to use stretching to prepare for a game. But does this make a difference? Th...

Introduction To Statistics And Data Analysis

Let the function f be defined by y=fx, where x and fx are real numbers. Find f2, f-3, fk, and fk2-1. fx=2x+4

College Algebra (MindTap Course List)

The purpose of an independent-measures t-test is to determine whether the mean difference obtained between two ...

Research Methods for the Behavioral Sciences (MindTap Course List)

For each measurement find: a. the degree of precision. b. the value that is equal to or less than the range of ...

Mathematics For Machine Technology

Finding Slope and Concavity In Exercises 9-18, find dy/dx and d2y/dx2, and find the slope b and concavity (if p...

Calculus (MindTap Course List)

In Exercises 1-10, find the cardinal numbers of the set in each given pair to determine whether the sets are eq...

Mathematics: A Practical Odyssey

Construct a stem-and-leaf display for the following data.

Essentials Of Statistics For Business & Economics

Average Traffic Spacing The headway h is the average time between vehicles. On a highway carrying an average of...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

(a) Use the substitution v = dy/dt to solve (13) for v in termsof y. Assuming that the velocity of the rocket a...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

In 4—9, express each statement using -, O-, or -notation. 9. n36n2(n31)n3 for every integer n2 . (Use notati...

Discrete Mathematics With Applications

Price Comparison of Smoothie Blenders. A personal fitness company produces both a deluxe and a standard model o...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)