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 6.3.64
Chapter 6, Problem 6.3.64

Let f(x) = {x   if 0≤x≤2
      2x−2  if 2<x≤5
      −2x+18 if 5<x≤6. 


Find the volume of the solid formed when the region bounded by the graph of f, the x-axis, and the line x=6 is revolved about the x-axis.
fig

Verified step by step guidance
1
Identify the piecewise function \( f(x) \) defined as: \[ f(x) = \begin{cases} x & \text{if } 0 \leq x \leq 2 \\ 2x - 2 & \text{if } 2 < x \leq 5 \\ -2x + 18 & \text{if } 5 < x \leq 6 \end{cases} \] This function describes the height of the region above the x-axis for each interval.
Recall that the volume of the solid formed by revolving the region bounded by \( f(x) \), the x-axis, and the vertical line \( x = 6 \) about the x-axis can be found using the disk method. The volume \( V \) is given by the integral: \[ V = \pi \int_a^b [f(x)]^2 \, dx \] where \( a = 0 \) and \( b = 6 \) in this problem.
Since \( f(x) \) is piecewise, split the integral into three parts corresponding to the intervals: \[ V = \pi \left( \int_0^2 (x)^2 \, dx + \int_2^5 (2x - 2)^2 \, dx + \int_5^6 (-2x + 18)^2 \, dx \right) \]
Set up each integral explicitly: - For \( 0 \leq x \leq 2 \): \[ \int_0^2 x^2 \, dx \] - For \( 2 < x \leq 5 \): \[ \int_2^5 (2x - 2)^2 \, dx \] - For \( 5 < x \leq 6 \): \[ \int_5^6 (-2x + 18)^2 \, dx \]
Evaluate each integral separately and then sum the results. Finally, multiply the sum by \( \pi \) to find the total volume of the solid formed by revolving the region about the x-axis.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7m
Was this helpful?

Key Concepts

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

Piecewise Functions

A piecewise function is defined by different expressions over distinct intervals of the domain. Understanding how to interpret and work with each piece separately is essential, especially when integrating or analyzing the function over its entire domain.
Recommended video:
05:36
Piecewise Functions

Volume of Solids of Revolution

The volume of a solid formed by revolving a region around an axis can be found using methods like the disk or washer method. This involves integrating the cross-sectional area, typically π[f(x)]², along the axis of revolution.
Recommended video:
04:48
Finding Volume Using Disks

Definite Integration

Definite integration calculates the accumulation of quantities, such as area or volume, over an interval. For solids of revolution, it is used to sum the volumes of infinitesimally thin disks or washers across the given bounds.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

Find the area of the region described in the following exercises.


The region in the first quadrant bounded by y=x^2/3 and y=4

71
views
Textbook Question

Assume f and g are continuous, with f(x) ≥ g(x) ≥ 0 on [a, b]. The region bounded by the graphs of f and g and the lines x=a and x=b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.

46
views
Textbook Question

Leaky Bucket A 1-kg bucket resting on the ground contains 3 kg of water. How much work is required to raise the bucket vertically a distance of 10 m if water leaks out of the bucket at a constant rate of 1/5 kg/m? Assume the weight of the rope used to raise the bucket is negligible. (Hint: Use the definition of work, W = ∫a^bF(y) dy, where F is the variable force required to lift an object along a vertical line from y=a to y=b.)

72
views
Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.


x=2−secy,x=2,y=π/3, and y=0; about x=2

163
views
Textbook Question

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.


x =2

48
views
Textbook Question

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.


ρ(x) = {1 if 0≤x≤2 {2 if 2<x≤3

37
views