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

The gap between two curves

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

  1. The line y = x meets the parabola y = x squared. What are the intersection points?

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

  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. Once found, what do the crossing x values become in your 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. The line y = 6x and the parabola y = x squared enclose a region. What is its area?

    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. What is the area enclosed by the line y = x and the parabola y = x squared?

    • 1/6
    • 1/3
    • 1/2
    • 2/3

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

    Circle one:   True   False

  10. Priya says: if the curves cross inside the interval, split the integral at the crossing point and add the separate areas. Is Priya right?

    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. (0, 0) and (1, 1) · Setting x equal to x squared gives x = 0 or x = 1, so the curves meet at (0, 0) and (1, 1).
  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. 36 · The curves cross at x = 0 and x = 6, and integrating 6x minus x squared between them gives 108 minus 72 = 36.
  6. Integral of x minus x cubed from 0 to 1 · Testing 1 over 2 shows the line higher, so it goes first.
  7. 1 over 6 · The line leads from 0 to 1, and 1 over 2 minus 1 over 3 is 1 over 6.
  8. 1/6 · Between x = 0 and x = 1 the line is on top, and integrating x minus x squared gives 1/2 minus 1/3 = 1/6.
  9. True · Splitting keeps top minus bottom correct in every piece.
  10. True · Priya is right. Splitting at each crossing keeps top minus bottom correct in every piece so nothing cancels.
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