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(a) Find a positive continuous function f such that the area under the graph of f from 0 to t is A ( t ) = t 3 for all t > 0. (b) A solid is generated by rotating about the x -axis the region under the curve y = f ( x ), where f is a positive function and x … 0. The volume generated by the part of the curve from x = 0 to x = b is b 2 for all b > 0. Find the function f .

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Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636
Chapter 5, Problem 1P
Textbook Problem
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(a) Find a positive continuous function f such that the area under the graph of f from 0 to t is A(t) = t3 for all t > 0.

(b) A solid is generated by rotating about the x-axis the region under the curve y = f(x), where f is a positive function and x … 0. The volume generated by the part of the curve from x = 0 to x = b is b2 for all b > 0. Find the function f.

(a)

To determine

To find: The positive continuous function f.

Explanation of Solution

Given information:

Area under the graph of f from 0 to t is A(t)=t3 for all t>0.

Calculation:

Show the Equation of the area under the graph as follows.

0tA(t)dt=t3 (1)

Apply the Fundamental Theorem of Calculus, Part 1 as follows.

If f is continuous on [a,b], then the function g is defined by

g(x)=axf(t)dtaxb

Function g is continuous on [a,b] and differentiable on (a,b), and g(x)=f(x)

(b)

To determine

To find: The function f.

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Chapter 5 Solutions

Single Variable Calculus
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Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - The graphs of two functions are shown with the...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Use calculus to find the area of the triangle with...Ch. 5.1 - Use calculus to find the area of the triangle with...Ch. 5.1 - Evaluate the integral and interpret it as the area...Ch. 5.1 - Evaluate the integral and interpret it as the area...Ch. 5.1 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Use a computer algebra system to find the exact...Ch. 5.1 - Sketch the region in the xy-plane defined by the...Ch. 5.1 - Racing cars driven by Chris and Kelly are side by...Ch. 5.1 - The widths (in meters) of a kidney-shaped swimming...Ch. 5.1 - A cross-section of an airplane wing is shown....Ch. 5.1 - If the birth rate of a population is b(t) = 2200 +...Ch. 5.1 - In Example 5, we modeled a measles pathogenesis...Ch. 5.1 - The rates at which rain fell, in inches per hour,...Ch. 5.1 - Two cars, A and B, start side by side and...Ch. 5.1 - The figure shows graphs of the marginal revenue...Ch. 5.1 - The curve with equation y2 = x2(x + 3) is called...Ch. 5.1 - Find the area of the region bounded by the...Ch. 5.1 - Find the number b such that the line y = b divides...Ch. 5.1 - (a) Find the number a such that the line x = a...Ch. 5.1 - Find the values of c such that the area of the...Ch. 5.1 - Suppose that 0 c /2. 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(b)...Ch. 5 - Suppose that you push a book across a 6-meter-long...Ch. 5 - (a) What is the average value of a function f on...Ch. 5 - Find the area of the region bounded by the given...Ch. 5 - Find the area of the region bounded by the given...Ch. 5 - Find the area of the region bounded by the given...Ch. 5 - Find the area of the region bounded by the given...Ch. 5 - Find the area of the region bounded by the given...Ch. 5 - Find the area of the region bounded by the given...Ch. 5 - Find the volume of the solid obtained by rotating...Ch. 5 - Find the volume of the solid obtained by rotating...Ch. 5 - Find the volume of the solid obtained by rotating...Ch. 5 - Find the volume of the solid obtained by rotating...Ch. 5 - Find the volume of the solid obtained by rotating...Ch. 5 - Set up, but do not evaluate, an integral for the...Ch. 5 - Set up, but do not evaluate, an integral for the...Ch. 5 - Set up, but do not evaluate, an integral for the...Ch. 5 - Find the volumes of the solids obtained by...Ch. 5 - Let R be the region in the first quadrant bounded...Ch. 5 - Let R be the region bounded by the curves y =...Ch. 5 - Let R be the region bounded by the curves y = 1 ...Ch. 5 - Each integral represents the volume of a solid....Ch. 5 - Each integral represents the volume of a solid....Ch. 5 - Each integral represents the volume of a solid....Ch. 5 - Each integral represents the volume of a solid....Ch. 5 - The base of a solid is a circular disk with radius...Ch. 5 - The base of a solid is the region bounded by the...Ch. 5 - The height of a monument is 20 m. 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