Skip to main content
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 6.3.8d
Chapter 6, Problem 6.3.8d

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.

Verified step by step guidance
1
Step 1: Understand the problem. The region R is bounded by the graphs of y = 1 + √x, x = 4, and y = 1. The goal is to find the volume of the solid formed when this region is revolved about the y-axis using the washer method.
Step 2: Express x in terms of y for the curve y = 1 + √x. Rearrange the equation to isolate √x: √x = y - 1. Then square both sides to get x = (y - 1)^2.
Step 3: Determine the bounds for y. The region starts at y = 1 (the horizontal line) and ends at y = 1 + √4 = 3 (since x = 4). Thus, the bounds for y are from 1 to 3.
Step 4: Set up the washer method formula for the volume. The volume of the solid is given by the integral: V = π ∫[y₁ to y₂] [(Outer radius)^2 - (Inner radius)^2] dy. Here, the outer radius is x = (y - 1)^2, and the inner radius is x = 4.
Step 5: Write the integral. Substitute the bounds and radii into the formula: V = π ∫[1 to 3] [(y - 1)^4 - 4^2] dy. This integral represents the volume of the solid formed by revolving the region R about the y-axis.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

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 y-axis, which means the volume of the solid can be calculated using methods such as the disk or washer method, depending on the shape of the cross-sections.
Recommended video:
04:48
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 are washers (disks with holes). It involves integrating the area of the outer radius minus the area of the inner radius, typically expressed as π(R^2 - r^2), where R and r are the outer and inner radii, respectively.
Recommended video:
07:33
Euler's Method

Definite Integral

A definite integral is a mathematical expression that calculates the accumulation of quantities, such as area or volume, over a specific interval. In this context, the definite integral will be set up to evaluate the volume of the solid formed by revolving region R around the y-axis, integrating from the lower to the upper bounds of the region.
Recommended video:
05:43
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 x-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.

1
views
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

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.


c. When will the tank be full?

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?

Textbook Question

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

c. The Spokane River flows out of Lake Coeur d’Alene, which contains approximately 0.67mi³ of water. Determine the percentage of Lake Coeur d’Alene’s volume that flows through Spokane in April and June.