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Volumes by cylindrical shells practice problems

Solids of revolution built from nested cylinders: 2πx f(x) dx. 63 problems with worked solutions; in 63 of them every equation is proved by a computer algebra system.

12 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 5.31

easy
The region under \( \displaystyle y = 4 - x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{2} 2 \pi x \left(4 - x^{2}\right)\, dx = 8 \pi \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( 8 \pi \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.32

easy
The region under \( \displaystyle y = e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{1} 2 \pi x e^{- x^{2}}\, dx = - \frac{\pi}{e} + \pi \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( - \frac{\pi}{e} + \pi \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.33

easy
The region under \( \displaystyle y = x^{3} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{2} 2 \pi x^{4}\, dx = \frac{64 \pi}{5} \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( \frac{64 \pi}{5} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.34

easy
The region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{2} 2 \pi x^{\frac{3}{2}}\, dx = \frac{16 \sqrt{2} \pi}{5} \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( \frac{16 \sqrt{2} \pi}{5} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.36

easy
The region under \( \displaystyle y = x \left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{3} 2 \pi x^{2} \left(3 - x\right)\, dx = \frac{27 \pi}{2} \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( \frac{27 \pi}{2} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.41

easy
The region under \( \displaystyle y = 2 \sqrt{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{1} 4 \pi x^{\frac{3}{2}}\, dx = \frac{8 \pi}{5} \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( \frac{8 \pi}{5} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.42

easy
The region under \( \displaystyle y = 3 x^{3} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{3} 6 \pi x^{4}\, dx = \frac{1458 \pi}{5} \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( \frac{1458 \pi}{5} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.45

easy
The region under \( \displaystyle y = 3 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{3} 6 \pi x e^{- x^{2}}\, dx = - \frac{3 \pi}{e^{9}} + 3 \pi \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( - \frac{3 \pi}{e^{9}} + 3 \pi \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.79

easy
The region under \( \displaystyle y = \sin{\left(x^{2} \right)} \) for \( \displaystyle 0 \le x \le sqrt(pi) \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{\sqrt{\pi}} 2 \pi x \sin{\left(x^{2} \right)}\, dx = 2 \pi \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( 2 \pi \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.82

easy
The region under \( \displaystyle y = 3 x \left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{3} 6 \pi x^{2} \left(3 - x\right)\, dx = \frac{81 \pi}{2} \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( \frac{81 \pi}{2} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.88

easy
The region under \( \displaystyle y = 3 \sin{\left(x^{2} \right)} \) for \( \displaystyle 0 \le x \le sqrt(pi) \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{\sqrt{\pi}} 6 \pi x \sin{\left(x^{2} \right)}\, dx = 6 \pi \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( 6 \pi \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 5.89

easy
The region under \( \displaystyle y = 8 - 2 x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
  1. A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.
    Reviewed
  2. \[ \int\limits_{0}^{2} 2 \pi x \left(8 - 2 x^{2}\right)\, dx = 16 \pi \]
    V = ∫ 2πx f(x) dx.✓ Proved
Answer \( 16 \pi \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More volumes by cylindrical shells practice

The region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.35easy✓ Nihil obstat
The region under \( \displaystyle y = x^{3} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.37easy✓ Nihil obstat
The region under \( \displaystyle y = x^{2} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.38easy✓ Nihil obstat
The region under \( \displaystyle y = e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.39easy✓ Nihil obstat
The region under \( \displaystyle y = x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.40easy✓ Nihil obstat
The region under \( \displaystyle y = 2 x^{2} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.43easy✓ Nihil obstat
The region under \( \displaystyle y = \frac{1}{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.44easy✓ Every equation proved
The region under \( \displaystyle y = 3 \sqrt{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.78easy✓ Nihil obstat
The region under \( \displaystyle y = \frac{3}{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.80easy✓ Every equation proved
The region under \( \displaystyle y = 2 \sqrt{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.81easy✓ Nihil obstat
The region under \( \displaystyle y = \frac{1}{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.83easy✓ Every equation proved
The region under \( \displaystyle y = 3 \sqrt{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.84easy✓ Nihil obstat
The region under \( \displaystyle y = 3 x^{3} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.85easy✓ Nihil obstat
The region under \( \displaystyle y = x^{2} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.86easy✓ Nihil obstat
The region under \( \displaystyle y = 2 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.87easy✓ Nihil obstat
The region under \( \displaystyle y = 2 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.90easy✓ Nihil obstat
The region under \( \displaystyle y = 3 x^{2} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.91easy✓ Nihil obstat
The region under \( \displaystyle y = 2 x \left(3 - x\right) \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.92easy✓ Nihil obstat
The region under \( \displaystyle y = \frac{3}{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.102easy✓ Every equation proved
The region under \( \displaystyle y = \frac{2}{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.103easy✓ Every equation proved
The region under \( \displaystyle y = 2 x^{2} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.104easy✓ Nihil obstat
The region under \( \displaystyle y = 2 \sin{\left(x^{2} \right)} \) for \( \displaystyle 0 \le x \le sqrt(pi) \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.105easy✓ Nihil obstat
The region under \( \displaystyle y = e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.106easy✓ Nihil obstat
The region under \( \displaystyle y = 3 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.107easy✓ Nihil obstat
The region under \( \displaystyle y = 2 x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.108easy✓ Nihil obstat
The region under \( \displaystyle y = 3 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.109easy✓ Nihil obstat
The region under \( \displaystyle y = 12 - 3 x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.126easy✓ Nihil obstat
The region under \( \displaystyle y = 2 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.127easy✓ Nihil obstat
The region under \( \displaystyle y = 2 \sqrt{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.128easy✓ Nihil obstat
The region under \( \displaystyle y = x^{3} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.129easy✓ Nihil obstat
The region under \( \displaystyle y = 3 x^{3} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.130easy✓ Nihil obstat
The region under \( \displaystyle y = 3 x^{2} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.131easy✓ Nihil obstat
The region under \( \displaystyle y = \frac{2}{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.132easy✓ Every equation proved
The region under \( \displaystyle y = e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.133easy✓ Nihil obstat
The region under \( \displaystyle y = 2 x^{2} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.150easy✓ Nihil obstat
The region under \( \displaystyle y = 2 x^{3} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.151easy✓ Nihil obstat
The region under \( \displaystyle y = x^{2} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.152easy✓ Nihil obstat
The region under \( \displaystyle y = 3 \sqrt{x} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.153easy✓ Nihil obstat
The region under \( \displaystyle y = 3 \sqrt{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.154easy✓ Nihil obstat
The region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 3 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.155easy✓ Nihil obstat
The region under \( \displaystyle y = x^{3} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.156easy✓ Nihil obstat
The region under \( \displaystyle y = 3 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.157easy✓ Nihil obstat
The region under \( \displaystyle y = \frac{3}{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.174easy✓ Every equation proved
The region under \( \displaystyle y = 2 x^{3} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.175easy✓ Nihil obstat
The region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.176easy✓ Nihil obstat
The region under \( \displaystyle y = \frac{2}{x} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.177easy✓ Every equation proved
The region under \( \displaystyle y = 2 x^{3} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.178easy✓ Nihil obstat
The region under \( \displaystyle y = 2 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.179easy✓ Nihil obstat
The region under \( \displaystyle y = 2 \sqrt{x} \) for \( \displaystyle 0 \le x \le 4 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.180easy✓ Nihil obstat
The region under \( \displaystyle y = \frac{1}{x} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.181easy✓ Every equation proved
The region under \( \displaystyle y = 3 x^{2} \) for \( \displaystyle 0 \le x \le 1 \) is revolved about the y-axis. Use shells to find the volume.
Problem 5.198easy✓ Nihil obstat