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Area and surface area with parametric curves practice problems

The area under a parametric curve, and the surface area when it is rotated about the x-axis. 29 problems with worked solutions; in 29 of them every equation is proved by a computer algebra system.

10 worked problems

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

Problem 8.253

hard
Find the area of the surface made by rotating \( \displaystyle x = 4 t,\ y = 3 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
  1. S = ∫ 2πy √(x′(t)² + y′(t)²) dt.
    Reviewed
  2. \[ \left(\frac{d}{d t} 3 t\right)^{2} + \left(\frac{d}{d t} 4 t\right)^{2} = 25 \]
    x′² + y′², simplified.✓ Proved
  3. \[ \int\limits_{0}^{1} 30 \pi t\, dt = 15 \pi \]
    Integrate.✓ Proved
Answer \( 15 \pi \)

✓ Nihil obstat Lines: 2 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 8.259

hard
Find the area of the surface made by rotating \( \displaystyle x = 2 \cos{\left(t \right)},\ y = 2 \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \pi \), about the \( \displaystyle x \)-axis.
  1. S = ∫ 2πy √(x′(t)² + y′(t)²) dt.
    Reviewed
  2. \[ \left(\frac{d}{d t} 2 \sin{\left(t \right)}\right)^{2} + \left(\frac{d}{d t} 2 \cos{\left(t \right)}\right)^{2} = 4 \]
    x′² + y′², simplified.✓ Proved
  3. \[ \int\limits_{0}^{\pi} 8 \pi \sin{\left(t \right)}\, dt = 16 \pi \]
    Integrate.✓ Proved
Answer \( 16 \pi \)

✓ Nihil obstat Lines: 2 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 8.260

hard
Find the area of the surface made by rotating \( \displaystyle x = t^{3},\ y = t^{3} \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
  1. S = ∫ 2πy √(x′(t)² + y′(t)²) dt.
    Reviewed
  2. \[ 2 \left(\frac{d}{d t} t^{3}\right)^{2} = 18 t^{4} \]
    x′² + y′², simplified.✓ Proved
  3. \[ \int\limits_{0}^{1} 6 \sqrt{2} \pi t^{5}\, dt = \sqrt{2} \pi \]
    Integrate.✓ Proved
Answer \( \sqrt{2} \pi \)

✓ Nihil obstat Lines: 2 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 8.337

hard
Find the area of the surface made by rotating \( \displaystyle x = \cos{\left(t \right)},\ y = \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \pi \), about the \( \displaystyle x \)-axis.
  1. S = ∫ 2πy √(x′(t)² + y′(t)²) dt.
    Reviewed
  2. \[ \left(\frac{d}{d t} \sin{\left(t \right)}\right)^{2} + \left(\frac{d}{d t} \cos{\left(t \right)}\right)^{2} = 1 \]
    x′² + y′², simplified.✓ Proved
  3. \[ \int\limits_{0}^{\pi} 2 \pi \sin{\left(t \right)}\, dt = 4 \pi \]
    Integrate.✓ Proved
Answer \( 4 \pi \)

✓ Nihil obstat Lines: 2 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 8.388

hard
Find the area of the surface made by rotating \( \displaystyle x = 3 t,\ y = t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
  1. S = ∫ 2πy √(x′(t)² + y′(t)²) dt.
    Reviewed
  2. \[ \left(\frac{d}{d t} t\right)^{2} + \left(\frac{d}{d t} 3 t\right)^{2} = 10 \]
    x′² + y′², simplified.✓ Proved
  3. \[ \int\limits_{0}^{1} 2 \sqrt{10} \pi t\, dt = \sqrt{10} \pi \]
    Integrate.✓ Proved
Answer \( \sqrt{10} \pi \)

✓ Nihil obstat Lines: 2 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 8.447

hard
Find the area of the surface made by rotating \( \displaystyle x = t,\ y = 2 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
  1. S = ∫ 2πy √(x′(t)² + y′(t)²) dt.
    Reviewed
  2. \[ \left(\frac{d}{d t} t\right)^{2} + \left(\frac{d}{d t} 2 t\right)^{2} = 5 \]
    x′² + y′², simplified.✓ Proved
  3. \[ \int\limits_{0}^{1} 4 \sqrt{5} \pi t\, dt = 2 \sqrt{5} \pi \]
    Integrate.✓ Proved
Answer \( 2 \sqrt{5} \pi \)

✓ Nihil obstat Lines: 2 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 8.449

hard
Find the area of the surface made by rotating \( \displaystyle x = t,\ y = t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
  1. S = ∫ 2πy √(x′(t)² + y′(t)²) dt.
    Reviewed
  2. \[ 2 \left(\frac{d}{d t} t\right)^{2} = 2 \]
    x′² + y′², simplified.✓ Proved
  3. \[ \int\limits_{0}^{1} 2 \sqrt{2} \pi t\, dt = \sqrt{2} \pi \]
    Integrate.✓ Proved
Answer \( \sqrt{2} \pi \)

✓ Nihil obstat Lines: 2 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 8.250

medium
Find the area under the curve \( \displaystyle x = t^{3},\ y = t^{2} \) for \( \displaystyle 0 \le t \le 1 \) (above the \( \displaystyle x \)-axis).
  1. x increases with t here, so A = ∫ y dx = ∫ y(t) x′(t) dt.
    Reviewed
  2. \[ t^{2} \frac{d}{d t} t^{3} = 3 t^{4} \]
    y·x′(t).✓ Proved
  3. \[ \int\limits_{0}^{1} 3 t^{4}\, dt = \frac{3}{5} \]
    Integrate.✓ Proved
Answer \( \frac{3}{5} \)

✓ Nihil obstat Lines: 2 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 8.251

medium
Find the area under the curve \( \displaystyle x = t,\ y = t^{2} + 1 \) for \( \displaystyle 0 \le t \le 1 \) (above the \( \displaystyle x \)-axis).
  1. x increases with t here, so A = ∫ y dx = ∫ y(t) x′(t) dt.
    Reviewed
  2. \[ \left(t^{2} + 1\right) \frac{d}{d t} t = t^{2} + 1 \]
    y·x′(t).✓ Proved
  3. \[ \int\limits_{0}^{1} \left(t^{2} + 1\right)\, dt = \frac{4}{3} \]
    Integrate.✓ Proved
Answer \( \frac{4}{3} \)

✓ Nihil obstat Lines: 2 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 8.252

medium
Find the area under the curve \( \displaystyle x = 2 t + 1,\ y = t^{2} \) for \( \displaystyle 0 \le t \le 1 \) (above the \( \displaystyle x \)-axis).
  1. x increases with t here, so A = ∫ y dx = ∫ y(t) x′(t) dt.
    Reviewed
  2. \[ t^{2} \frac{d}{d t} \left(2 t + 1\right) = 2 t^{2} \]
    y·x′(t).✓ Proved
  3. \[ \int\limits_{0}^{1} 2 t^{2}\, dt = \frac{2}{3} \]
    Integrate.✓ Proved
Answer \( \frac{2}{3} \)

✓ Nihil obstat Lines: 2 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 area and surface area with parametric curves practice

Find the area under the curve \( \displaystyle x = t^{2},\ y = t^{3} \) for \( \displaystyle 0 \le t \le 1 \) (above the \( \displaystyle x \)-axis).
Problem 8.255medium✓ Nihil obstat
Find the area under the curve \( \displaystyle x = 2 t + 1,\ y = t^{2} \) for \( \displaystyle 0 \le t \le 2 \) (above the \( \displaystyle x \)-axis).
Problem 8.256medium✓ Nihil obstat
Find the area under the curve \( \displaystyle x = t^{3},\ y = t^{2} \) for \( \displaystyle 0 \le t \le 2 \) (above the \( \displaystyle x \)-axis).
Problem 8.257medium✓ Nihil obstat
Find the area under the curve \( \displaystyle x = t^{2},\ y = t^{3} \) for \( \displaystyle 0 \le t \le 2 \) (above the \( \displaystyle x \)-axis).
Problem 8.258medium✓ Nihil obstat
Find the area under the curve \( \displaystyle x = t,\ y = t^{2} + 1 \) for \( \displaystyle 0 \le t \le 2 \) (above the \( \displaystyle x \)-axis).
Problem 8.262medium✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 3 t,\ y = 3 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.254hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 4 \cos{\left(t \right)},\ y = 4 \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \pi \), about the \( \displaystyle x \)-axis.
Problem 8.261hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 2 t,\ y = 4 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.263hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 3 \cos{\left(t \right)},\ y = 3 \sin{\left(t \right)} \), \( \displaystyle 0 \le t \le \pi \), about the \( \displaystyle x \)-axis.
Problem 8.264hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 2 t,\ y = 3 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.335hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 4 t,\ y = 2 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.336hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = t,\ y = 4 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.448hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 4 t,\ y = 4 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.450hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = t,\ y = 3 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.451hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 4 t,\ y = t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.452hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 3 t,\ y = 2 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.453hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 2 t,\ y = t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.474hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 3 t,\ y = 4 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.475hard✓ Nihil obstat
Find the area of the surface made by rotating \( \displaystyle x = 2 t,\ y = 2 t \), \( \displaystyle 0 \le t \le 1 \), about the \( \displaystyle x \)-axis.
Problem 8.476hard✓ Nihil obstat