Areas Between Two Curves · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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The gap between two curves

Mathematics · Calculus & Analysis · ages 17-18
Name ______________________   Date ____________

  1. The area between y = 9 and y = x² is restricted to 1 <= x <= 2. Which limits describe this restricted area?

    • -3 and 3
    • 1 and 2
    • 0 and 3
    • -2 and 2

  2. The line y equals x meets the parabola y equals x squared. What are the points?

    • (0, 0) and (2, 4)
    • (0, 0) and (1, 1)
    • (0, 0) only

  3. For a region enclosed solely by two curves, what do the outer intersection x-values give in the area integral?

    • The limits of integration
    • Numbers you add to the final answer
    • Values you can ignore from here on

  4. The line y equals x plus 2 crosses y equals x squared twice. Give the smaller crossing x value.

    Answer: ______________

  5. On two consecutive intervals, f - g does not change sign within either interval. Its signed integrals are 3 and -5. What is the total geometric area between the graphs over both intervals?

    Answer: ______________

  6. From 0 to 1, which setup gives the area between y equals x and y equals x cubed?

    • Integral of x cubed minus x from 0 to 1
    • Integral of x minus x cubed from 0 to 1
    • Integral of x plus x cubed from 0 to 1

  7. What is the area enclosed by y equals x and y equals x squared?

    • 1 over 3
    • 1 over 2
    • 1 over 6

  8. The integral of x² - 4 from 0 to 1 is -11/3. What is the geometric area between y = x² - 4 and the x-axis on that interval?

    • -11/3
    • 4/3
    • 11/3
    • 5/3

  9. Priya says a crossing inside the interval means split there and add the separate areas. Priya is right.

    Circle one:   True   False

  10. For the area between y = x² and y = 0 on [-1, 1], splitting at x = 0 is required because the curves exchange their upper and lower positions there.

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

The gap between two curves W1-mt_lDBZmnLK1b-s1

  1. 1 and 2 · The supplied vertical boundaries are x = 1 and x = 2, even though the curves meet elsewhere.
  2. (0, 0) and (1, 1) · Setting x equal to x squared gives 0 or 1, hence those two points.
  3. The limits of integration · Crossings mark where the gap starts and ends, so they are the limits.
  4. -1 · Solving x squared equals x plus 2 gives 2 or negative 1.
  5. 8 · There is no cancellation within either interval. Their geometric areas are the magnitudes 3 and 5, giving 8.
  6. Integral of x minus x cubed from 0 to 1 · Testing 1 over 2 shows that the line is higher, so subtract the cubic function from the line.
  7. 1 over 6 · The line is higher from 0 to 1, and 1 over 2 minus 1 over 3 is 1 over 6.
  8. 11/3 · The curve stays below the axis. Geometric area uses 0 - (x² - 4), reversing the signed result to 11/3.
  9. True · Splitting keeps top minus bottom correct in every piece.
  10. False · The square is nonnegative on both sides of zero. The curves touch but do not exchange order.
Worksheet · LightMySky