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Area between curves practice problems

Integrate top minus bottom between the points where the curves cross. 70 problems with worked solutions; in 70 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 5.1

medium Video
Find the area of the region bounded by \( \displaystyle y = - x^{2} + 4 x + 3 \) and \( \displaystyle y = 3 x + 1 \).
  1. \[ \left(2 - x\right) \left(x + 1\right) = - x^{2} + x + 2 \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = -1 and x = 2; between them y = -x**2 + 4*x + 3 is on top.
    Reviewed
  3. \[ \int\limits_{-1}^{2} \left(- x^{2} + x + 2\right)\, dx = \frac{9}{2} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{9}{2} \)
The curves y = -x² + 4x + 3 and y = 3x + 1 between x = -1 and x = 2. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 9/2.

✓ 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 5.2

medium Video
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 14 x - 11 \) and \( \displaystyle y = 2 x - 2 \).
  1. \[ \left(9 - 3 x\right) \left(x - 1\right) = - 3 x^{2} + 12 x - 9 \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = 1 and x = 3; between them y = -3*x**2 + 14*x - 11 is on top.
    Reviewed
  3. \[ \int\limits_{1}^{3} \left(- 3 x^{2} + 12 x - 9\right)\, dx = 4 \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( 4 \)
The curves y = -3x² + 14x - 11 and y = 2x - 2 between x = 1 and x = 3. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 4.

✓ 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 5.3

medium Video
Find the area of the region bounded by \( \displaystyle y = - x^{2} - x + 10 \) and \( \displaystyle y = 2 - 3 x \).
  1. \[ \left(4 - x\right) \left(x + 2\right) = - x^{2} + 2 x + 8 \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = -2 and x = 4; between them y = -x**2 - x + 10 is on top.
    Reviewed
  3. \[ \int\limits_{-2}^{4} \left(- x^{2} + 2 x + 8\right)\, dx = 36 \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( 36 \)
The curves y = -x² - x + 10 and y = 2 - 3x between x = -2 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 36.

✓ 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 5.5

medium Video
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 1 \) and \( \displaystyle y = 2 x - 1 \).
  1. \[ - x \left(x + 2\right) = - x^{2} - 2 x \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = -2 and x = 0; between them y = -x**2 - 1 is on top.
    Reviewed
  3. \[ \int\limits_{-2}^{0} \left(- x^{2} - 2 x\right)\, dx = \frac{4}{3} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{4}{3} \)
The curves y = -x² - 1 and y = 2x - 1 between x = -2 and x = 0. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 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 5.9

medium Video
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 6 x - 3 \) and \( \displaystyle y = - 2 x - 3 \).
  1. \[ - 2 x \left(x - 4\right) = - 2 x^{2} + 8 x \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = 0 and x = 4; between them y = -2*x**2 + 6*x - 3 is on top.
    Reviewed
  3. \[ \int\limits_{0}^{4} \left(- 2 x^{2} + 8 x\right)\, dx = \frac{64}{3} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{64}{3} \)
The curves y = -2x² + 6x - 3 and y = -2x - 3 between x = 0 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 64/3.

✓ 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 5.67

medium Video
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 5 x - 1 \) and \( \displaystyle y = - 2 x - 1 \).
  1. \[ - x \left(x + 3\right) = - x^{2} - 3 x \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = -3 and x = 0; between them y = -x**2 - 5*x - 1 is on top.
    Reviewed
  3. \[ \int\limits_{-3}^{0} \left(- x^{2} - 3 x\right)\, dx = \frac{9}{2} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{9}{2} \)
The curves y = -x² - 5x - 1 and y = -2x - 1 between x = -3 and x = 0. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 9/2.

✓ 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 5.73

medium Video
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 9 x + 7 \) and \( \displaystyle y = 3 x - 1 \).
  1. \[ \left(8 - 2 x\right) \left(x + 1\right) = - 2 x^{2} + 6 x + 8 \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = -1 and x = 4; between them y = -2*x**2 + 9*x + 7 is on top.
    Reviewed
  3. \[ \int\limits_{-1}^{4} \left(- 2 x^{2} + 6 x + 8\right)\, dx = \frac{125}{3} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{125}{3} \)
The curves y = -2x² + 9x + 7 and y = 3x - 1 between x = -1 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 125/3.

✓ 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 5.75

medium Video
Find the area of the region bounded by \( \displaystyle y = 3 - 2 x^{2} \) and \( \displaystyle y = 2 x - 1 \).
  1. \[ \left(2 - 2 x\right) \left(x + 2\right) = - 2 x^{2} - 2 x + 4 \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = -2 and x = 1; between them y = 3 - 2*x**2 is on top.
    Reviewed
  3. \[ \int\limits_{-2}^{1} \left(- 2 x^{2} - 2 x + 4\right)\, dx = 9 \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( 9 \)
The curves y = 3 - 2x² and y = 2x - 1 between x = -2 and x = 1. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 9.

✓ 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 5.167

medium Video
Find the area of the region bounded by \( \displaystyle y = - x^{2} + 2 x + 10 \) and \( \displaystyle y = x - 2 \).
  1. \[ \left(4 - x\right) \left(x + 3\right) = - x^{2} + x + 12 \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = -3 and x = 4; between them y = -x**2 + 2*x + 10 is on top.
    Reviewed
  3. \[ \int\limits_{-3}^{4} \left(- x^{2} + x + 12\right)\, dx = \frac{343}{6} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{343}{6} \)
The curves y = -x² + 2x + 10 and y = x - 2 between x = -3 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 343/6.

✓ 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 5.173

medium Video
Find the area of the region bounded by \( \displaystyle y = 15 - 3 x^{2} \) and \( \displaystyle y = - 3 x - 3 \).
  1. \[ \left(9 - 3 x\right) \left(x + 2\right) = - 3 x^{2} + 3 x + 18 \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = -2 and x = 3; between them y = 15 - 3*x**2 is on top.
    Reviewed
  3. \[ \int\limits_{-2}^{3} \left(- 3 x^{2} + 3 x + 18\right)\, dx = \frac{125}{2} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{125}{2} \)
The curves y = 15 - 3x² and y = -3x - 3 between x = -2 and x = 3. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 125/2.

✓ 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 5.192

medium Video
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 3 x - 1 \) and \( \displaystyle y = x - 1 \).
  1. \[ - 2 x \left(x - 1\right) = - 2 x^{2} + 2 x \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = 0 and x = 1; between them y = -2*x**2 + 3*x - 1 is on top.
    Reviewed
  3. \[ \int\limits_{0}^{1} \left(- 2 x^{2} + 2 x\right)\, dx = \frac{1}{3} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{1}{3} \)
The curves y = -2x² + 3x - 1 and y = x - 1 between x = 0 and x = 1. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 1/3.

✓ 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 5.195

medium Video
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 11 x + 3 \) and \( \displaystyle y = 3 x + 3 \).
  1. \[ - 2 x \left(x - 4\right) = - 2 x^{2} + 8 x \]
    Set the curves equal: top − bottom, factored.✓ Proved
  2. They cross at x = 0 and x = 4; between them y = -2*x**2 + 11*x + 3 is on top.
    Reviewed
  3. \[ \int\limits_{0}^{4} \left(- 2 x^{2} + 8 x\right)\, dx = \frac{64}{3} \]
    Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{64}{3} \)
The curves y = -2x² + 11x + 3 and y = 3x + 3 between x = 0 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 64/3.

✓ 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 between curves practice

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 10 x - 1 \) and \( \displaystyle y = - 2 x - 1 \).
Problem 5.4medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 5 x + 24 \) and \( \displaystyle y = 3 x \).
Problem 5.6medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + x + 14 \) and \( \displaystyle y = - 3 x - 2 \).
Problem 5.7medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} + x + 1 \) and \( \displaystyle y = 3 x + 1 \).
Problem 5.8medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} + x + 2 \) and \( \displaystyle y = x - 2 \).
Problem 5.10medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 10 x - 5 \) and \( \displaystyle y = x + 1 \).
Problem 5.11medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - 3 x - 1 \) and \( \displaystyle y = - x - 1 \).
Problem 5.12medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 13 x - 11 \) and \( \displaystyle y = 1 - 2 x \).
Problem 5.13medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 17 x - 16 \) and \( \displaystyle y = 2 x + 2 \).
Problem 5.14medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 7 x - 8 \) and \( \displaystyle y = - 3 x \).
Problem 5.15medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + x + 17 \) and \( \displaystyle y = 1 - 3 x \).
Problem 5.61medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 3 x + 15 \) and \( \displaystyle y = - x - 1 \).
Problem 5.62medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 13 x - 17 \) and \( \displaystyle y = x - 1 \).
Problem 5.63medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 3 x - 6 \) and \( \displaystyle y = - 3 x - 2 \).
Problem 5.64medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 11 x - 9 \) and \( \displaystyle y = x + 3 \).
Problem 5.65medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 16 x - 11 \) and \( \displaystyle y = x + 1 \).
Problem 5.66medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 6 x + 9 \) and \( \displaystyle y = 2 x + 3 \).
Problem 5.68medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 15 x - 21 \) and \( \displaystyle y = x + 3 \).
Problem 5.69medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + x + 15 \) and \( \displaystyle y = - 3 x - 1 \).
Problem 5.70medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 13 x - 12 \) and \( \displaystyle y = x - 3 \).
Problem 5.71medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 2 x - 3 \) and \( \displaystyle y = x - 3 \).
Problem 5.72medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + x + 3 \) and \( \displaystyle y = 3 x + 3 \).
Problem 5.74medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 13 x - 23 \) and \( \displaystyle y = 1 - x \).
Problem 5.94medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 13 x - 12 \) and \( \displaystyle y = - x - 3 \).
Problem 5.95medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 6 x - 9 \) and \( \displaystyle y = - x - 3 \).
Problem 5.96medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 18 x - 21 \) and \( \displaystyle y = 3 x - 3 \).
Problem 5.97medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 17 x - 26 \) and \( \displaystyle y = - x - 2 \).
Problem 5.98medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 12 x - 5 \) and \( \displaystyle y = 1 - 3 x \).
Problem 5.99medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 2 x + 21 \) and \( \displaystyle y = 3 - x \).
Problem 5.100medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - 3 x + 3 \) and \( \displaystyle y = 1 - 3 x \).
Problem 5.101medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - x + 16 \) and \( \displaystyle y = 2 x - 2 \).
Problem 5.118medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 4 x + 4 \) and \( \displaystyle y = x - 2 \).
Problem 5.119medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} + 7 x - 9 \) and \( \displaystyle y = x - 1 \).
Problem 5.120medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 2 x + 1 \) and \( \displaystyle y = 2 x - 1 \).
Problem 5.121medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 4 x - 3 \) and \( \displaystyle y = - 3 x - 3 \).
Problem 5.122medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - 3 x \) and \( \displaystyle y = - 3 x - 2 \).
Problem 5.123medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - x + 25 \) and \( \displaystyle y = - x - 2 \).
Problem 5.124medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 6 x - 8 \) and \( \displaystyle y = - 3 x - 2 \).
Problem 5.125medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 2 x + 7 \) and \( \displaystyle y = 1 - x \).
Problem 5.142medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} + 4 x \) and \( \displaystyle y = 2 x - 3 \).
Problem 5.143medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - 3 x - 4 \) and \( \displaystyle y = 3 x \).
Problem 5.144medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - x + 15 \) and \( \displaystyle y = 3 - 3 x \).
Problem 5.145medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - 4 x - 1 \) and \( \displaystyle y = 2 x + 3 \).
Problem 5.146medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 8 x + 6 \) and \( \displaystyle y = - 2 x - 3 \).
Problem 5.147medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 4 x \) and \( \displaystyle y = - 2 x - 3 \).
Problem 5.148medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 5 x + 10 \) and \( \displaystyle y = 2 - x \).
Problem 5.149medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - x + 7 \) and \( \displaystyle y = 2 x + 1 \).
Problem 5.166medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 2 x \) and \( \displaystyle y = - 3 x - 2 \).
Problem 5.168medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 12 x - 27 \) and \( \displaystyle y = - 2 x - 3 \).
Problem 5.169medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 6 x - 1 \) and \( \displaystyle y = 3 x - 1 \).
Problem 5.170medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 2 x + 16 \) and \( \displaystyle y = - x - 2 \).
Problem 5.171medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 11 x + 1 \) and \( \displaystyle y = 1 - x \).
Problem 5.172medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 17 x - 15 \) and \( \displaystyle y = 2 x + 3 \).
Problem 5.190medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} - 4 x + 3 \) and \( \displaystyle y = - 3 x - 3 \).
Problem 5.191medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 5 x + 19 \) and \( \displaystyle y = x + 3 \).
Problem 5.193medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - x^{2} + 3 x + 4 \) and \( \displaystyle y = x + 1 \).
Problem 5.194medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 3 x + 9 \) and \( \displaystyle y = 3 x \).
Problem 5.196medium Video✓ Every equation proved
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 13 x - 11 \) and \( \displaystyle y = 3 x + 1 \).
Problem 5.197medium Video✓ Every equation proved