Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.AAE.20a
Chapter 7, Problem 7.AAE.20a

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.

Verified step by step guidance
1
Identify the function and the interval: The curve is given by \(y = \frac{1}{2\sqrt{x}}\) and the interval is from \(x = \frac{1}{4}\) to \(x = 4\).
Recall the formula for the volume of a solid of revolution about the x-axis: \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\), where \(f(x)\) is the function being revolved.
Square the function inside the integral: \(\left( \frac{1}{2\sqrt{x}} \right)^{2} = \frac{1}{4x}\).
Set up the integral for the volume: \(V = \pi \int_{\frac{1}{4}}^{4} \frac{1}{4x} \, dx\).
Evaluate the integral by integrating \(\frac{1}{4x}\) with respect to \(x\) over the interval \(\left[ \frac{1}{4}, 4 \right]\), then multiply the result by \(\pi\) to find the volume.

Verified video answer for a similar problem:

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

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 formed when a plane region is rotated about a line (axis), creating a three-dimensional object. The volume of such solids can be found using integral calculus by summing infinitesimal disks or washers perpendicular to the axis of rotation.
Recommended video:
04:48
Finding Volume Using Disks

Disk Method

The disk method calculates the volume of a solid of revolution by slicing it into thin circular disks perpendicular to the axis. Each disk's volume is approximated by π(radius)^2 times thickness, and integrating these volumes over the interval gives the total volume.
Recommended video:
06:30
Disk Method Using y-Axis

Definite Integration

Definite integration computes the exact accumulation of quantities, such as area or volume, over a specific interval. In this problem, integrating the squared function from x=1/4 to x=4 calculates the total volume generated by revolving the region around the x-axis.
Recommended video:
Guided course
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

Textbook Question

Find the limits in Exercises 1–6.

3. lim(x→0⁺) (cox(√x))^(1/x)

Textbook Question

Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?

Textbook Question

7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?

Textbook Question

19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.

Textbook Question

Find the limits in Exercises 1–6.

1. lim(b→1⁻) ∫(from 0 to b) dx/√(1-x²)

1
views