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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.

Verified step by step guidance
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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

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 y-axis to form a solid of revolution whose cross sections are washers.


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

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from A units to 2A units is greater than the cost of increasing production from 2A units to 3A units.

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Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

Textbook Question

Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.


d. More generally, if the riders’ speeds are v(t)=A(t+1)² and u(t)=B(t+1)² and the distance between the towns is D, what conditions on A, B, and D must be met to ensure that the riders will pass each other?

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Textbook Question

Compressing and stretching a spring Suppose a force of 30 N is required to stretch and hold a spring 0.2 m from its equilibrium position.

d. How much additional work is required to stretch the spring 0.2m if it has already been stretched 0.2m from its equilibrium position?

Textbook Question

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function

v(t)={3t if 0≤t<2060 if 20≤t<45240−4t if t≥45v(t)= \(\begin{cases}\)3 t & \(\text\) { if } 0 \(\leq\) t<20 \\ 60 & \(\text\) { if } 20 \(\leq\) t<45 \\ 240-4 t & \(\text\) { if } t \(\geq\) 45\(\end{cases}\)

where is measured in seconds and v has units of m/s. 

d. What is the position of the automobile when t=75?