Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
14th Edition
ISBN: 9780134438986
Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher: PEARSON
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Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Chapter 6.4, Problem 19E
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Calculate the surface area generated by revolving the curve about y axis.

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Chapter 6 Solutions

Thomas' Calculus (14th Edition)

Ch. 6.1 - Prob. 1ECh. 6.1 - Find the volumes of the solids in Exercises...Ch. 6.1 - Find the volumes of the solids in Exercises...Ch. 6.1 - Find the volumes of the solids in Exercises...Ch. 6.1 - Find the volumes of the solids in Exercises...Ch. 6.1 - Find the volumes of the solids in Exercises...Ch. 6.1 - Find the volumes of the solids in Exercises...Ch. 6.1 - Find the volumes of the solids in Exercises...Ch. 6.1 - Find the volumes of the solids in Exercises...Ch. 6.1 - Find the volumes of the solids in Exercises...
Ch. 6.1 - Find the volume of the given right tetrahedron....Ch. 6.1 - Prob. 12ECh. 6.1 - A twisted solid A square of side length s lies in...Ch. 6.1 - Prob. 14ECh. 6.1 - Intersection of two half-cylinders Two...Ch. 6.1 - Gasoline in a tank A gasoline tank is in the shape...Ch. 6.1 - Prob. 17ECh. 6.1 - Prob. 18ECh. 6.1 - Prob. 19ECh. 6.1 - Prob. 20ECh. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - In Exercises 31 and 32, find the volume of the...Ch. 6.1 - In Exercises 31 and 32, find the volume of the...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Prob. 35ECh. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - Find the volumes of the solids generated by...Ch. 6.1 - In Exercises 47-50, find the volume of the solid...Ch. 6.1 - Prob. 44ECh. 6.1 - In Exercises 47-50, find the volume of the solid...Ch. 6.1 - In Exercises 47-50, find the volume of the solid...Ch. 6.1 - In Exercises 51 and 52, find the volume of the...Ch. 6.1 - In Exercises 51 and 52, find the volume of the...Ch. 6.1 - Find the volume of the solid generated by...Ch. 6.1 - Find the volume of the solid generated by...Ch. 6.1 - Find the volume of the solid generated by...Ch. 6.1 - By integration, find the volume of the solid...Ch. 6.1 - The volume of a torus The disk x2 + y2 ≤ a2 is...Ch. 6.1 - Prob. 54ECh. 6.1 - Prob. 55ECh. 6.1 - Prob. 56ECh. 6.1 - Volume of a hemisphere Derive the formula V =...Ch. 6.1 - Designing a plumb bob Having been asked to design...Ch. 6.1 - Designing a wok You are designing a wok frying pan...Ch. 6.1 - Max-min The arch y = sin x, 0 ≤ x ≤ π, is revolved...Ch. 6.1 - Prob. 61ECh. 6.1 - Prob. 62ECh. 6.2 - In Exercises 1–6, use the shell method to find the...Ch. 6.2 - In Exercises 1–6, use the shell method to find the...Ch. 6.2 - In Exercises 1–6, use the shell method to find the...Ch. 6.2 - In Exercises 1–6, use the shell method to find the...Ch. 6.2 - In Exercises 1–6, use the shell method to find the...Ch. 6.2 - In Exercises 1–6, use the shell method to find the...Ch. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - Prob. 9ECh. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - Prob. 13ECh. 6.2 - Prob. 14ECh. 6.2 - Prob. 15ECh. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - Prob. 17ECh. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - Prob. 19ECh. 6.2 - Prob. 20ECh. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - Use the shell method to find the volumes of the...Ch. 6.2 - In Exercises 23–26, use the shell method to find...Ch. 6.2 - In Exercises 23–26, use the shell method to find...Ch. 6.2 - In Exercises 23–26, use the shell method to find...Ch. 6.2 - In Exercises 23–26, use the shell method to find...Ch. 6.2 - In Exercises 27 and 28, use the shell method to...Ch. 6.2 - Prob. 28ECh. 6.2 - For some regions, both the washer and shell...Ch. 6.2 - Prob. 30ECh. 6.2 - Prob. 31ECh. 6.2 - Prob. 32ECh. 6.2 - Prob. 33ECh. 6.2 - In Exercises 31–36, find the volumes of the solids...Ch. 6.2 - Prob. 35ECh. 6.2 - In Exercises 31–36, find the volumes of the solids...Ch. 6.2 - Prob. 37ECh. 6.2 - The region in the first quadrant that is bounded...Ch. 6.2 - The region shown here is to be revolved about the...Ch. 6.2 - Prob. 40ECh. 6.2 - Prob. 41ECh. 6.2 - Prob. 42ECh. 6.2 - Prob. 43ECh. 6.2 - Prob. 44ECh. 6.2 - Consider the region R bounded by the graphs of y =...Ch. 6.2 - Consider the region R given in Exercise 45. If the...Ch. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Prob. 5ECh. 6.3 - Prob. 6ECh. 6.3 - Prob. 7ECh. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Find the lengths of the curves in Exercises 1–16....Ch. 6.3 - Prob. 13ECh. 6.3 - Prob. 14ECh. 6.3 - In Exercises 17-24, do the following. Set up an...Ch. 6.3 - Prob. 16ECh. 6.3 - Prob. 17ECh. 6.3 - Prob. 18ECh. 6.3 - In Exercises 17-24, do the following. Set up an...Ch. 6.3 - In Exercises 17-24, do the following. Set up an...Ch. 6.3 - Find a curve with a positive derivative through...Ch. 6.3 - Prob. 22ECh. 6.3 - Find the length of the curve from x = 0 to x =...Ch. 6.3 - The length of an astroid The graph of the equation...Ch. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - If 9x2 = y(y − 3)2, that Ch. 6.3 - Prob. 28ECh. 6.3 - Prob. 29ECh. 6.3 - Prob. 30ECh. 6.3 - Prob. 31ECh. 6.3 - Prob. 32ECh. 6.3 - Find the arc length function for the graph of f(x)...Ch. 6.3 - Prob. 34ECh. 6.4 - In Exercises 1-8: Set up an integral for the area...Ch. 6.4 - In Exercises 1-8: Set up an integral for the area...Ch. 6.4 - Prob. 3ECh. 6.4 - In Exercises 1-8: Set up an integral for the area...Ch. 6.4 - In Exercises 1-8: Set up an integral for the area...Ch. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Find the lateral (side) surface area of the cone...Ch. 6.4 - Find the lateral surface area of the cone...Ch. 6.4 - Prob. 11ECh. 6.4 - Prob. 12ECh. 6.4 - Find the areas of the surfaces generated by...Ch. 6.4 - Prob. 14ECh. 6.4 - Find the areas of the surfaces generated by...Ch. 6.4 - Find the areas of the surfaces generated by...Ch. 6.4 - Find the areas of the surfaces generated by...Ch. 6.4 - Find the areas of the surfaces generated by...Ch. 6.4 - Find the areas of the surfaces generated by...Ch. 6.4 - Prob. 20ECh. 6.4 - Prob. 21ECh. 6.4 - Prob. 22ECh. 6.4 - Prob. 23ECh. 6.4 - Prob. 24ECh. 6.4 - Prob. 25ECh. 6.4 - Prob. 26ECh. 6.4 - Prob. 27ECh. 6.4 - Prob. 28ECh. 6.4 - Prob. 29ECh. 6.4 - Prob. 30ECh. 6.4 - Prob. 31ECh. 6.4 - The surface of an astroid Find the area of the...Ch. 6.5 - Prob. 1ECh. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Stretching a spring A spring has a natural length...Ch. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Subway car springs It takes a force of 21,714 lb...Ch. 6.5 - Bathroom scale A bathroom scale is compressed 1/16...Ch. 6.5 - Lifting a rope A mountain climber is about to haul...Ch. 6.5 - Leaky sandbag A bag of sand originally weighing...Ch. 6.5 - Prob. 11ECh. 6.5 - Prob. 12ECh. 6.5 - Leaky bucket Assume the bucket in Example 4 is...Ch. 6.5 - Prob. 14ECh. 6.5 - Pumping water The rectangular tank shown here,...Ch. 6.5 - Emptying a cistern The rectangular cistern...Ch. 6.5 - Pumping oil How much work would it take to pump...Ch. 6.5 - Prob. 18ECh. 6.5 - Emptying a tank A vertical right-circular...Ch. 6.5 - Prob. 20ECh. 6.5 - The graph of y = x2 on 0 ≤ x ≤ 2 is revolved about...Ch. 6.5 - A right-circular cylindrical tank of height 10 ft...Ch. 6.5 - Prob. 23ECh. 6.5 - Prob. 24ECh. 6.5 - Prob. 25ECh. 6.5 - Prob. 26ECh. 6.5 - In Exercises 26–30, use the result of Exercise...Ch. 6.5 - Prob. 28ECh. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.5 - Prob. 31ECh. 6.5 - Water tower Your town has decided to drill a well...Ch. 6.5 - Prob. 33ECh. 6.5 - Forcing electrons together Two electrons r meters...Ch. 6.5 - Triangular plate Calculate the fluid force on one...Ch. 6.5 - Triangular plate Calculate the fluid force on one...Ch. 6.5 - Prob. 37ECh. 6.5 - Prob. 38ECh. 6.5 - Triangular plate The isosceles triangular plate...Ch. 6.5 - Prob. 40ECh. 6.5 - Prob. 41ECh. 6.5 - Prob. 42ECh. 6.5 - Prob. 43ECh. 6.5 - Prob. 44ECh. 6.5 - Prob. 45ECh. 6.5 - Prob. 46ECh. 6.5 - Prob. 47ECh. 6.5 - Prob. 48ECh. 6.5 - Prob. 49ECh. 6.5 - Watering trough The vertical ends of a watering...Ch. 6.6 - In Exercises 1–6, find the mass M and center of...Ch. 6.6 - In Exercises 1–6, find the mass M and center of...Ch. 6.6 - In Exercises 1–6, find the mass M and center of...Ch. 6.6 - In Exercises 1–6, find the mass M and center of...Ch. 6.6 - Prob. 5ECh. 6.6 - In Exercises 1–6, find the mass M and center of...Ch. 6.6 - In Exercises 7–20, find the center of mass of a...Ch. 6.6 - In Exercises 7–20, find the center of mass of a...Ch. 6.6 - Prob. 9ECh. 6.6 - Prob. 10ECh. 6.6 - Prob. 11ECh. 6.6 - Prob. 12ECh. 6.6 - Prob. 13ECh. 6.6 - Prob. 14ECh. 6.6 - Prob. 15ECh. 6.6 - Prob. 16ECh. 6.6 - In Exercises 7–20, find the center of mass of a...Ch. 6.6 - Prob. 18ECh. 6.6 - Prob. 19ECh. 6.6 - Prob. 20ECh. 6.6 - Prob. 21ECh. 6.6 - Prob. 22ECh. 6.6 - The region bounded by the curves and the lines x...Ch. 6.6 - Prob. 24ECh. 6.6 - Prob. 25ECh. 6.6 - Prob. 26ECh. 6.6 - Prob. 27ECh. 6.6 - Prob. 28ECh. 6.6 - Prob. 29ECh. 6.6 - Prob. 30ECh. 6.6 - Prob. 31ECh. 6.6 - Prob. 32ECh. 6.6 - Prob. 33ECh. 6.6 - Prob. 34ECh. 6.6 - In Exercises 37-40, find the centroid of the thin...Ch. 6.6 - Prob. 36ECh. 6.6 - In Exercises 37-40, find the centroid of the thin...Ch. 6.6 - Prob. 38ECh. 6.6 - Prob. 39ECh. 6.6 - Prob. 40ECh. 6.6 - Prob. 41ECh. 6.6 - Use a theorem of Pappus to find the volume...Ch. 6.6 - Prob. 43ECh. 6.6 - Prob. 44ECh. 6.6 - Use Pappus’s Theorem for surface area and the fact...Ch. 6.6 - Prob. 46ECh. 6.6 - The area of the region R enclosed by the...Ch. 6.6 - As found in Example 8, the centroid of the region...Ch. 6.6 - Prob. 49ECh. 6.6 - Prob. 50ECh. 6.6 - Prob. 51ECh. 6.6 - Prob. 52ECh. 6 - Prob. 1GYRCh. 6 - How are the disk and washer methods for...Ch. 6 - Prob. 3GYRCh. 6 - How do you find the length of the graph of a...Ch. 6 - How do you define and calculate the area of the...Ch. 6 - Prob. 6GYRCh. 6 - What is a center of mass? What is a centroid? Ch. 6 - Prob. 8GYRCh. 6 - Prob. 9GYRCh. 6 - How do you locate the center of mass of a thin...Ch. 6 - Prob. 1PECh. 6 - Prob. 2PECh. 6 - Find the volumes of the solids in Exercises...Ch. 6 - Prob. 4PECh. 6 - Prob. 5PECh. 6 - Prob. 6PECh. 6 - Find the volumes of the solids in Exercises...Ch. 6 - Prob. 8PECh. 6 - Prob. 9PECh. 6 - Prob. 10PECh. 6 - Prob. 11PECh. 6 - Prob. 12PECh. 6 - Prob. 13PECh. 6 - Prob. 14PECh. 6 - Prob. 15PECh. 6 - Prob. 16PECh. 6 - Prob. 17PECh. 6 - Find the volumes of the solids in Exercises...Ch. 6 - Prob. 19PECh. 6 - Prob. 20PECh. 6 - Lengths of Curves Find the lengths of the curves...Ch. 6 - Prob. 22PECh. 6 - Prob. 23PECh. 6 - Prob. 24PECh. 6 - Prob. 25PECh. 6 - Prob. 26PECh. 6 - Prob. 27PECh. 6 - Prob. 28PECh. 6 - Prob. 29PECh. 6 - Prob. 30PECh. 6 - Prob. 31PECh. 6 - Pumping a reservoir (Continuation of Exercise 31.)...Ch. 6 - Prob. 33PECh. 6 - Pumping a cylindrical tank A storage tank is a...Ch. 6 - Prob. 35PECh. 6 - Prob. 36PECh. 6 - Prob. 37PECh. 6 - Prob. 38PECh. 6 - Prob. 39PECh. 6 - Prob. 40PECh. 6 - Prob. 41PECh. 6 - Prob. 42PECh. 6 - Prob. 43PECh. 6 - Prob. 44PECh. 6 - Prob. 45PECh. 6 - Prob. 46PECh. 6 - Prob. 1AAECh. 6 - Prob. 2AAECh. 6 - Prob. 3AAECh. 6 - Prob. 4AAECh. 6 - Prob. 5AAECh. 6 - Consider a right-circular cylinder of diameter 1....Ch. 6 - Prob. 7AAECh. 6 - Prob. 8AAECh. 6 - Prob. 9AAECh. 6 - Prob. 10AAECh. 6 - Prob. 11AAECh. 6 - Prob. 12AAECh. 6 - Prob. 13AAECh. 6 - Prob. 14AAECh. 6 - Prob. 15AAECh. 6 - Prob. 16AAECh. 6 - Prob. 17AAECh. 6 - Prob. 18AAE
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