Multivariable Calculus (looseleaf)
11th Edition
ISBN: 9781337275590
Author: Larson
Publisher: Cengage
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Textbook Question
Chapter 14.6, Problem 56E
Moments of Inertia In Exercises 53- 56, find
(a)
(b)
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Provethat
a) prove that for any irrational numbers there exists?
asequence of rational numbers Xn converg to S.
b) let S: RR be a sunctions-t.
f(x)=(x-1) arc tan (x), xe Q
3(x-1)
1+x²
x&Q
Show that lim f(x)= 0
14x
C) For any set A define the set -A=y
Q2: Find the interval and radius of convergence for the following series:
Σ
n=1
(-1)η-1
xn
n
8. Evaluate arctan x dx
a) xartanx
2
2
In(1 + x²) + C b) xartanx + 1½-3ln(1 + x²) + C c) xartanx + In(1 + x²) + C d)
(arctanx)²
+ C
2
9) Evaluate Inx³ dx
3
a) +C b) ln x² + C c)¾½ (lnx)² d) 3x(lnx − 1) + C
-
x
10) Determine which integral is obtained when the substitution x =
So¹² √1 - x²dx
sine is made in the integral
πT
π
π
a) √ sin cos e de b) √ cos² de c) c
Ꮎ Ꮎ
cos² 0 de c)
cos e de d) for cos² e de
πT
11. Evaluate tan³xdx
1
a) b) c) [1 - In 2]
2
2
c) [1 − In2] d)½½[1+ In 2]
Chapter 14 Solutions
Multivariable Calculus (looseleaf)
Ch. 14.1 - CONCEPT CHECK Iterated Integral Explain what is...Ch. 14.1 - Prob. 2ECh. 14.1 - Prob. 3ECh. 14.1 - Prob. 4ECh. 14.1 - Prob. 5ECh. 14.1 - Prob. 6ECh. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...Ch. 14.1 - Prob. 8ECh. 14.1 - Prob. 9ECh. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...
Ch. 14.1 - Prob. 11ECh. 14.1 - Prob. 12ECh. 14.1 - Prob. 13ECh. 14.1 - Prob. 14ECh. 14.1 - Prob. 15ECh. 14.1 - Prob. 16ECh. 14.1 - Evaluating an Iterated Integral In Exercises...Ch. 14.1 - Prob. 18ECh. 14.1 - Evaluating an Iterated Integral In Exercises...Ch. 14.1 - Prob. 20ECh. 14.1 - Prob. 21ECh. 14.1 - Prob. 22ECh. 14.1 - Evaluating an Iterated Integral In Exercises...Ch. 14.1 - Prob. 24ECh. 14.1 - Evaluating an Iterated Integral In Exercises...Ch. 14.1 - Prob. 26ECh. 14.1 - Evaluating an Iterated Integral In Exercises...Ch. 14.1 - Prob. 28ECh. 14.1 - Evaluating an Improper Iterated Integral In...Ch. 14.1 - Prob. 30ECh. 14.1 - Evaluating an Improper Iterated Integral In...Ch. 14.1 - Prob. 32ECh. 14.1 - Prob. 33ECh. 14.1 - Prob. 34ECh. 14.1 - Prob. 35ECh. 14.1 - Prob. 36ECh. 14.1 - Prob. 37ECh. 14.1 - Prob. 38ECh. 14.1 - Prob. 39ECh. 14.1 - Prob. 40ECh. 14.1 - Finding the Area of a Region In Exercises 37-42,...Ch. 14.1 - Finding the Area of a Region In Exercises 37-42,...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Prob. 44ECh. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Prob. 51ECh. 14.1 - Prob. 52ECh. 14.1 - Prob. 53ECh. 14.1 - Prob. 54ECh. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Prob. 56ECh. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Prob. 58ECh. 14.1 - Prob. 59ECh. 14.1 - Prob. 60ECh. 14.1 - Prob. 61ECh. 14.1 - Prob. 62ECh. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Prob. 64ECh. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Prob. 67ECh. 14.1 - Prob. 68ECh. 14.1 - Prob. 69ECh. 14.1 - Prob. 70ECh. 14.1 - Prob. 71ECh. 14.1 - Prob. 72ECh. 14.1 - Prob. 73ECh. 14.1 - Prob. 74ECh. 14.1 - Prob. 75ECh. 14.1 - Prob. 76ECh. 14.1 - Prob. 77ECh. 14.1 - Prob. 78ECh. 14.1 - Prob. 79ECh. 14.1 - Prob. 80ECh. 14.2 - Prob. 1ECh. 14.2 - Prob. 2ECh. 14.2 - Prob. 3ECh. 14.2 - Prob. 4ECh. 14.2 - Prob. 5ECh. 14.2 - Approximation In Exercises 3-6, approximate the...Ch. 14.2 - Prob. 7ECh. 14.2 - Prob. 8ECh. 14.2 - Prob. 9ECh. 14.2 - Prob. 10ECh. 14.2 - Prob. 11ECh. 14.2 - Prob. 12ECh. 14.2 - Prob. 13ECh. 14.2 - Prob. 14ECh. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - Evaluating a Double IntegralIn Exercises 1320, set...Ch. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.2 - Prob. 20ECh. 14.2 - Prob. 21ECh. 14.2 - Finding Volume In Exercises 21-26, use a double...Ch. 14.2 - Prob. 23ECh. 14.2 - Prob. 24ECh. 14.2 - Prob. 25ECh. 14.2 - Prob. 26ECh. 14.2 - Prob. 27ECh. 14.2 - Prob. 28ECh. 14.2 - Finding Volume In Exercises 29-34, set up and...Ch. 14.2 - Prob. 30ECh. 14.2 - Prob. 31ECh. 14.2 - Prob. 32ECh. 14.2 - Prob. 33ECh. 14.2 - Prob. 34ECh. 14.2 - Prob. 35ECh. 14.2 - Prob. 36ECh. 14.2 - Prob. 37ECh. 14.2 - Prob. 38ECh. 14.2 - Prob. 39ECh. 14.2 - Prob. 40ECh. 14.2 - Prob. 41ECh. 14.2 - Prob. 42ECh. 14.2 - Prob. 43ECh. 14.2 - Prob. 44ECh. 14.2 - Prob. 45ECh. 14.2 - Prob. 46ECh. 14.2 - Prob. 47ECh. 14.2 - Prob. 48ECh. 14.2 - Prob. 49ECh. 14.2 - Prob. 50ECh. 14.2 - Prob. 51ECh. 14.2 - Prob. 52ECh. 14.2 - Prob. 53ECh. 14.2 - Prob. 54ECh. 14.2 - Prob. 55ECh. 14.2 - Prob. 56ECh. 14.2 - Prob. 57ECh. 14.2 - Average Temperature The temperature in degrees...Ch. 14.2 - Prob. 59ECh. 14.2 - VolumeLet the plane region R be a unit circle and...Ch. 14.2 - Prob. 61ECh. 14.2 - Prob. 62ECh. 14.2 - Prob. 63ECh. 14.2 - Prob. 64ECh. 14.2 - Prob. 65ECh. 14.2 - Prob. 66ECh. 14.2 - Prob. 67ECh. 14.2 - Prob. 68ECh. 14.2 - Prob. 69ECh. 14.2 - Prob. 70ECh. 14.2 - Maximizing a Double Integral Determine the region...Ch. 14.2 - Minimizing a Double Integral Determine the region...Ch. 14.2 - Prob. 73ECh. 14.2 - Prob. 74ECh. 14.2 - Prob. 75ECh. 14.2 - Prob. 76ECh. 14.3 - CONCEPT CHECK Choosing a Coordinate System In...Ch. 14.3 - Prob. 2ECh. 14.3 - Prob. 3ECh. 14.3 - Prob. 4ECh. 14.3 - Prob. 5ECh. 14.3 - Prob. 6ECh. 14.3 - Prob. 7ECh. 14.3 - Prob. 8ECh. 14.3 - Prob. 9ECh. 14.3 - Prob. 10ECh. 14.3 - Prob. 11ECh. 14.3 - Prob. 12ECh. 14.3 - Prob. 13ECh. 14.3 - Prob. 14ECh. 14.3 - Evaluating a Double Integral in Exercises 9-16,...Ch. 14.3 - Prob. 16ECh. 14.3 - Prob. 17ECh. 14.3 - Prob. 18ECh. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Prob. 20ECh. 14.3 - Converting to Polar Coordinates In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Prob. 24ECh. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises 27...Ch. 14.3 - Converting to Polar Coordinates: In Exercises 27...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates In Exercises 2932,...Ch. 14.3 - Converting to Polar Coordinates In Exercises 2932,...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Prob. 33ECh. 14.3 - Prob. 34ECh. 14.3 - Prob. 35ECh. 14.3 - Prob. 36ECh. 14.3 - Prob. 37ECh. 14.3 - In Exercises 3338, use a double integral in polar...Ch. 14.3 - Volume Use a double integral in polar coordinates...Ch. 14.3 - Prob. 40ECh. 14.3 - Prob. 41ECh. 14.3 - Prob. 42ECh. 14.3 - Prob. 43ECh. 14.3 - Prob. 44ECh. 14.3 - Prob. 45ECh. 14.3 - AreaIn Exercises 4146, use a double integral to...Ch. 14.3 - Prob. 47ECh. 14.3 - Prob. 48ECh. 14.3 - Area: In Exercises 4752, sketch a graph of the...Ch. 14.3 - Area: In Exercises 4752, sketch a graph of the...Ch. 14.3 - Area: In Exercises 4752, sketch a graph of the...Ch. 14.3 - Prob. 52ECh. 14.3 - EXPLORING CONCEPTS Area Express the area of the...Ch. 14.3 - Prob. 54ECh. 14.3 - Prob. 55ECh. 14.3 - Prob. 56ECh. 14.3 - Volume Determine the diameter of a hole that is...Ch. 14.3 - Prob. 58ECh. 14.3 - Prob. 59ECh. 14.3 - Prob. 60ECh. 14.3 - Prob. 61ECh. 14.3 - Prob. 62ECh. 14.3 - Probability The value of the integral I=ex22dx Is...Ch. 14.3 - Prob. 64ECh. 14.3 - Prob. 65ECh. 14.3 - Prob. 66ECh. 14.3 - Prob. 67ECh. 14.3 - Prob. 68ECh. 14.4 - Mass of a Planar Lamina Explain when you should...Ch. 14.4 - Prob. 2ECh. 14.4 - Prob. 3ECh. 14.4 - Prob. 4ECh. 14.4 - Prob. 5ECh. 14.4 - Prob. 6ECh. 14.4 - Finding the Center of Mass In Exercises 7-10, find...Ch. 14.4 - Prob. 8ECh. 14.4 - Finding the Center of Mass In Exercises 7-10, find...Ch. 14.4 - Prob. 10ECh. 14.4 - Prob. 11ECh. 14.4 - Prob. 12ECh. 14.4 - Finding the Center of Mass In Exercises 1324, find...Ch. 14.4 - Finding the Center of Mass In Exercises 1324, find...Ch. 14.4 - Finding the Center of Mass In Exercises 1324, find...Ch. 14.4 - Prob. 16ECh. 14.4 - Finding the Center of Mass In Exercises 1324, find...Ch. 14.4 - Prob. 18ECh. 14.4 - Prob. 19ECh. 14.4 - Prob. 20ECh. 14.4 - Finding the Center of Mass In Exercises 1324, find...Ch. 14.4 - Prob. 22ECh. 14.4 - Prob. 23ECh. 14.4 - Prob. 24ECh. 14.4 - Prob. 25ECh. 14.4 - Prob. 26ECh. 14.4 - Finding the Center of Mass Using Technology In...Ch. 14.4 - Prob. 28ECh. 14.4 - Prob. 29ECh. 14.4 - Prob. 30ECh. 14.4 - Prob. 31ECh. 14.4 - Prob. 32ECh. 14.4 - Prob. 33ECh. 14.4 - Finding the Radius of Gyration About Each Axis in...Ch. 14.4 - Prob. 35ECh. 14.4 - Prob. 36ECh. 14.4 - Prob. 37ECh. 14.4 - Finding Moments of Inertia and Radii of Gyration...Ch. 14.4 - Prob. 39ECh. 14.4 - Prob. 40ECh. 14.4 - Prob. 41ECh. 14.4 - Prob. 42ECh. 14.4 - Prob. 43ECh. 14.4 - Prob. 44ECh. 14.4 - Prob. 45ECh. 14.4 - Prob. 46ECh. 14.4 - Prob. 47ECh. 14.4 - HOW DO YOU SEE IT? The center of mass of the...Ch. 14.4 - Proof Prove the following Theorem of Pappus: Let R...Ch. 14.5 - CONCEPT CHECK Surface Area What is the...Ch. 14.5 - Prob. 2ECh. 14.5 - Prob. 3ECh. 14.5 - Prob. 4ECh. 14.5 - Prob. 5ECh. 14.5 - Prob. 6ECh. 14.5 - Finding Surface AreaIn Exercises 316, find the...Ch. 14.5 - Prob. 8ECh. 14.5 - Prob. 9ECh. 14.5 - Prob. 10ECh. 14.5 - Finding Surface AreaIn Exercises 316, find the...Ch. 14.5 - Prob. 12ECh. 14.5 - Finding Surface AreaIn Exercises 316, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 316, find the...Ch. 14.5 - Prob. 15ECh. 14.5 - Prob. 16ECh. 14.5 - Prob. 17ECh. 14.5 - Prob. 18ECh. 14.5 - Finding Surface Area In Exercises 17-20, find the...Ch. 14.5 - Prob. 20ECh. 14.5 - Prob. 21ECh. 14.5 - Prob. 22ECh. 14.5 - Prob. 23ECh. 14.5 - Prob. 24ECh. 14.5 - Prob. 25ECh. 14.5 - Prob. 26ECh. 14.5 - Prob. 27ECh. 14.5 - Prob. 28ECh. 14.5 - Setting Up a Double IntegralIn Exercises 2730, set...Ch. 14.5 - Prob. 30ECh. 14.5 - Prob. 31ECh. 14.5 - HOW DO YOU SEE IT? Consider the surface...Ch. 14.5 - Prob. 33ECh. 14.5 - Prob. 34ECh. 14.5 - Product DesignA company produces a spherical...Ch. 14.5 - Prob. 36ECh. 14.5 - Surface Area Find the surface area of the solid of...Ch. 14.5 - Prob. 38ECh. 14.6 - CONCEPT CHECK Triple Integrals What does Q=QdV...Ch. 14.6 - Prob. 2ECh. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral Using...Ch. 14.6 - Evaluating a Triple Iterated Integral Using...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Prob. 14ECh. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Prob. 16ECh. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Prob. 18ECh. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Prob. 26ECh. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Changing the Order of Integration In Exercises...Ch. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 35 and 36, the...Ch. 14.6 - Orders of Integration In Exercises 35 and 36, the...Ch. 14.6 - Center of Mass In Exercises 37-40, find the mass...Ch. 14.6 - Prob. 38ECh. 14.6 - Center of Mass In Exercises 37-40, find the mass...Ch. 14.6 - Center of Mass In Exercises 37-40, find the mass...Ch. 14.6 - Center of Mass In Exercises 41 and 42, set up the...Ch. 14.6 - Prob. 42ECh. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Prob. 51ECh. 14.6 - Prob. 52ECh. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Prob. 54ECh. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Moments of Inertia In Exercises 57 and 58, verify...Ch. 14.6 - Moments of Inertia In Exercises 57 and 58, verify...Ch. 14.6 - Moments of Inertia In Exercises 59 and 60, set up...Ch. 14.6 - Moments of Inertia In Exercises 59 and 60, set up...Ch. 14.6 - Prob. 61ECh. 14.6 - Prob. 62ECh. 14.6 - Prob. 63ECh. 14.6 - Average Value In Exercises 63-66, find the average...Ch. 14.6 - Average Value In Exercises 63-66, find the average...Ch. 14.6 - Average Value In Exercises 63-66, find the average...Ch. 14.6 - Prob. 67ECh. 14.6 - Prob. 68ECh. 14.6 - Prob. 69ECh. 14.6 - HOW DO YOU SEE IT? Consider two solids of equal...Ch. 14.6 - Maximizing a Triple Integral Find the solid region...Ch. 14.6 - Find a Value Solve for a in the triple integral....Ch. 14.6 - PUTNAM EXAM CHALLENGE Evaluate limn0101...01cos2{...Ch. 14.7 - CONCEPT CHECK Volume Explain why triple integrals...Ch. 14.7 - Prob. 2ECh. 14.7 - Prob. 3ECh. 14.7 - Prob. 4ECh. 14.7 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.7 - Prob. 6ECh. 14.7 - Prob. 7ECh. 14.7 - Prob. 8ECh. 14.7 - Prob. 9ECh. 14.7 - Prob. 10ECh. 14.7 - Volume In Exercises 11-14, sketch the solid region...Ch. 14.7 - Prob. 12ECh. 14.7 - Prob. 13ECh. 14.7 - Prob. 14ECh. 14.7 - VolumeIn Exercises 1520, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Prob. 17ECh. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Prob. 19ECh. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Prob. 21ECh. 14.7 - Mass In Exercises 21 and 22, use cylindrical...Ch. 14.7 - Using Cylindrical Coordinates In Exercises 23-28,...Ch. 14.7 - Using Cylindrical Coordinates In Exercises 23-28,...Ch. 14.7 - Prob. 27ECh. 14.7 - Prob. 29ECh. 14.7 - Prob. 31ECh. 14.7 - Prob. 32ECh. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Prob. 34ECh. 14.7 - Prob. 35ECh. 14.7 - Prob. 36ECh. 14.7 - Center of Mass In Exercises 37 and 38, use...Ch. 14.7 - Prob. 38ECh. 14.7 - Prob. 39ECh. 14.7 - Prob. 40ECh. 14.7 - Prob. 41ECh. 14.7 - Prob. 43ECh. 14.7 - Prob. 44ECh. 14.7 - Prob. 45ECh. 14.7 - Prob. 46ECh. 14.7 - Prob. 47ECh. 14.8 - CONCEPT CHECK JacobianDescribe how to find the...Ch. 14.8 - Prob. 2ECh. 14.8 - Prob. 3ECh. 14.8 - Prob. 4ECh. 14.8 - Prob. 5ECh. 14.8 - Prob. 6ECh. 14.8 - Prob. 7ECh. 14.8 - Prob. 8ECh. 14.8 - Prob. 9ECh. 14.8 - Prob. 10ECh. 14.8 - Using a Transformation In Exercises 11-14, sketch...Ch. 14.8 - Prob. 12ECh. 14.8 - Prob. 13ECh. 14.8 - Prob. 14ECh. 14.8 - Prob. 15ECh. 14.8 - Prob. 16ECh. 14.8 - Prob. 17ECh. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Prob. 26ECh. 14.8 - Prob. 27ECh. 14.8 - Prob. 28ECh. 14.8 - Prob. 29ECh. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Prob. 31ECh. 14.8 - Prob. 32ECh. 14.8 - Prob. 33ECh. 14.8 - Prob. 34ECh. 14.8 - Prob. 35ECh. 14.8 - Prob. 36ECh. 14.8 - Prob. 37ECh. 14.8 - Prob. 38ECh. 14.8 - Prob. 39ECh. 14.8 - Prob. 40ECh. 14.8 - Prob. 41ECh. 14 - Evaluating an Integral In Exercises 1 and 2,...Ch. 14 - Prob. 2RECh. 14 - Evaluating an Integral In Exercises 3 - 6,...Ch. 14 - Prob. 4RECh. 14 - Prob. 5RECh. 14 - Prob. 6RECh. 14 - Prob. 7RECh. 14 - Prob. 8RECh. 14 - Prob. 9RECh. 14 - Prob. 10RECh. 14 - Prob. 11RECh. 14 - Prob. 12RECh. 14 - Changing the Order of Integration In Exercises...Ch. 14 - Prob. 14RECh. 14 - Prob. 15RECh. 14 - Prob. 16RECh. 14 - Prob. 17RECh. 14 - Prob. 18RECh. 14 - Prob. 19RECh. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Prob. 21RECh. 14 - Prob. 22RECh. 14 - Prob. 23RECh. 14 - Prob. 24RECh. 14 - Prob. 25RECh. 14 - Prob. 26RECh. 14 - Prob. 27RECh. 14 - Prob. 28RECh. 14 - Prob. 29RECh. 14 - Prob. 30RECh. 14 - AreaIn Exercises 31 and 32, sketch a graph of the...Ch. 14 - Prob. 32RECh. 14 - Prob. 33RECh. 14 - Converting to Polar Coordinates Write the sum of...Ch. 14 - Prob. 35RECh. 14 - Prob. 36RECh. 14 - Prob. 37RECh. 14 - Finding the Center of Mass In Exercises 37-40,...Ch. 14 - Prob. 39RECh. 14 - Prob. 40RECh. 14 - Prob. 41RECh. 14 - Finding Moments of Inertia and Radii of GyrationIn...Ch. 14 - Finding Surface AreaIn Exercises 4346, find the...Ch. 14 - Finding Surface AreaIn Exercises 4346, find the...Ch. 14 - Prob. 45RECh. 14 - Finding Surface AreaIn Exercises 4346, find the...Ch. 14 - Prob. 47RECh. 14 - Prob. 48RECh. 14 - Prob. 49RECh. 14 - Prob. 50RECh. 14 - Prob. 51RECh. 14 - Prob. 52RECh. 14 - Prob. 53RECh. 14 - Prob. 54RECh. 14 - Prob. 55RECh. 14 - Prob. 56RECh. 14 - Changing the Order of Integration In Exercises 57...Ch. 14 - Prob. 59RECh. 14 - Center of Mass In Exercises 59 and 60, find the...Ch. 14 - Prob. 61RECh. 14 - Prob. 62RECh. 14 - Prob. 63RECh. 14 - Prob. 64RECh. 14 - Prob. 65RECh. 14 - Prob. 66RECh. 14 - VolumeIn Exercises 67 and 68, use cylindrical...Ch. 14 - Prob. 68RECh. 14 - Prob. 69RECh. 14 - Prob. 70RECh. 14 - Prob. 71RECh. 14 - Prob. 72RECh. 14 - Finding a JcobianIn Exercises 7174, find the...Ch. 14 - Prob. 74RECh. 14 - Prob. 75RECh. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - Prob. 77RECh. 14 - Prob. 78RECh. 14 - VolumeFind the volume of the solid of intersection...Ch. 14 - Surface AreaLet a,b,c, and d be positive real...Ch. 14 - Using a Change of variables The figure shows the...Ch. 14 - ProofProve that limn0101xnyndxdy=0.Ch. 14 - Deriving a Sum Derive Eulers famous result that...Ch. 14 - Evaluating a Double IntegralEvaluate the integral...Ch. 14 - Evaluating Double IntegralsEvaluate the integrals...Ch. 14 - VolumeShow that the volume of a spherical block...Ch. 14 - Evaluating an IntegralIn Exercises 9 and 10,...Ch. 14 - Prob. 10PSCh. 14 - Prob. 11PSCh. 14 - Prob. 12PSCh. 14 - Prob. 14PSCh. 14 - Prob. 15PSCh. 14 - SprinklerConsider a circular lawn with a radius of...Ch. 14 - Volume The figure shows a solid bounded below by...
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- 12. Evaluate ſ √9-x2 -dx. x2 a) C 9-x2 √9-x2 - x2 b) C - x x arcsin ½-½ c) C + √9 - x² + arcsin x d) C + √9-x2 x2 13. Find the indefinite integral S cos³30 √sin 30 dᎾ . 2√√sin 30 (5+sin²30) √sin 30 (3+sin²30) a) C+ √sin 30(5-sin²30) b) C + c) C + 5 5 5 10 d) C + 2√√sin 30 (3-sin²30) 2√√sin 30 (5-sin²30) e) C + 5 15 14. Find the indefinite integral ( sin³ 4xcos 44xdx. a) C+ (7-5cos24x)cos54x b) C (7-5cos24x)cos54x (7-5cos24x)cos54x - 140 c) C - 120 140 d) C+ (7-5cos24x)cos54x e) C (7-5cos24x)cos54x 4 4 15. Find the indefinite integral S 2x2 dx. ex - a) C+ (x²+2x+2)ex b) C (x² + 2x + 2)e-* d) C2(x²+2x+2)e¯* e) C + 2(x² + 2x + 2)e¯* - c) C2x(x²+2x+2)e¯*arrow_forward4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4arrow_forward1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.arrow_forward
- 2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?arrow_forwardQuestion 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.arrow_forwardPlease find all values of x.arrow_forward
- 3. Consider the initial value problem 9y" +12y' + 4y = 0, y(0) = a>0: y′(0) = −1. Solve the problem and find the value of a such that the solution of the initial value problem is always positive.arrow_forward5. Euler's equation. Determine the values of a for which all solutions of the equation 5 x²y" + axy' + y = 0 that have the form (A + B log x) x* or Ax¹¹ + Bä” tend to zero as a approaches 0.arrow_forward4. Problem on variable change. The purpose of this problem is to perform an appropriate change of variables in order to reduce the problem to a second-order equation with constant coefficients. ty" + (t² − 1)y'′ + t³y = 0, 0arrow_forward4. Some psychologists contend that the number of facts of a certain type that are remembered after t hours is given by f(t)== 90t 951-90 Find the rate at which the number of facts remembered is changing after 1 hour and after 10 hours. Interpret.arrow_forward12:05 MA S 58 58. If f(x) = ci.metaproxy.org 25 2xon [0, 10] and n is a positive integer, then there is some Riemann sum Sthat equals the exact area under the graph of ƒ from x = Oto x = 10. 59. If the area under the graph of fon [a, b] is equal to both the left sum L, and the right sum Rfor some positive integer n, then fis constant on [a, b]. 60. If ƒ is a decreasing function on [a, b], then the area under the graph of fis greater than the left sum Land less than the right sum R₂, for any positive integer n. Problems 61 and 62 refer to the following figure showing two parcels of land along a river: River Parcel 2 Parcel 1 h(x) 500 ft 1,000 ft. Figure for 61 and 62 61. You want to purchase both parcels of land shown in the figure and make a quick check on their combined area. There is no equation for the river frontage, so you use the average of the left and right sums of rectangles covering the area. The 1,000-foot baseline is divided into 10 equal parts. At the end of each…arrow_forwardIf 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/minarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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