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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.3.7d
Chapter 6, Problem 6.3.7d

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

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Step 1: Identify the boundaries of the region R. The region is bounded by the curves y = 1 + √x, x = 4, and y = 1. The curve y = 1 + √x is the upper boundary, and y = 1 is the lower boundary.
Step 2: Recognize that the solid is formed by revolving the region R about the x-axis. The cross-sections of the solid are washers, which means the volume can be calculated using the washer method.
Step 3: Write the formula for the volume using the washer method. The volume is given by the integral: V = ∫[a to b] π[(outer radius)^2 - (inner radius)^2] dx. Here, the outer radius is the distance from the x-axis to the curve y = 1 + √x, and the inner radius is the distance from the x-axis to the line y = 1.
Step 4: Determine the limits of integration. The region R spans from x = 0 to x = 4, so the limits of integration are a = 0 and b = 4.
Step 5: Substitute the expressions for the outer and inner radii into the formula. The outer radius is y = 1 + √x, and the inner radius is y = 1. The integral becomes: V = ∫[0 to 4] π[(1 + √x)^2 - (1)^2] dx.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solid of Revolution

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional area around an axis. In this case, the region R is revolved around the x-axis, forming a solid. Understanding this concept is crucial for visualizing the shape and determining how to calculate its volume.
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Finding Volume Using Disks

Washer Method

The washer method is a technique used to find the volume of a solid of revolution when the cross-sections perpendicular to the axis of rotation are washers (disks with holes). This method involves integrating the area of the washers, which is calculated as the difference between the outer and inner radii squared, multiplied by π, over the interval of integration.
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Euler's Method

Definite Integral

A definite integral represents the accumulation of quantities, such as area or volume, over a specific interval. In this context, it is used to calculate the volume of the solid formed by revolving region R around the x-axis. The limits of integration correspond to the bounds of the region, and the integrand is derived from the washer method.
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Definition of the Definite Integral
Related Practice
Textbook Question

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.


e. How far has the racer traveled when it reaches a speed of 178 ft/s?

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Textbook Question

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Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


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Textbook Question

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e. Describe the position of the object relative to its initial position after 8 seconds.

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Textbook Question

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Textbook Question

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

d. What is the displacement of the object over the interval [0,5]?

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. 


d. Let f(x)=12x^2.. The area of the surface generated when the graph of f on [−4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.

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