The Definite Integral and the Area Under a Curve · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

From rectangles to exact area

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

  1. Which working correctly evaluates the integral from 1 to 3 of 2x dx?

    • [x²] from 1 to 3 = 9 - 1 = 8
    • [2x²] from 1 to 3 = 18 - 2 = 16
    • [x²] from 1 to 3 = 3 - 1 = 2
    • [x] from 1 to 3 = 3 - 1 = 2

  2. Evaluate the integral from 1 to 3 of 2x dx using square-bracket notation.

    Answer: ______________

  3. Which working correctly evaluates the integral from 1 to 3 of 2x dx?

    • Square bracket 2x squared, from 1 to 3, giving 18 minus 2
    • Square bracket x squared, from 1 to 3, giving 9 minus 1, which is 8
    • Square bracket x, from 1 to 3, giving 3 minus 1

  4. A definite integral is a single number, while an indefinite integral is a family of functions.

    Circle one:   True   False

  5. Tom says the integral from 0 to 2 of x squared minus 1 equals the area there. Tom is right.

    Circle one:   True   False

  6. Tom says the integral from 0 to 2 of (x² - 1) dx equals the area between the curve and the x-axis. Is Tom right?

    Circle one:   True   False

  7. Two left-endpoint rectangles estimate the area under y = x² from 0 to 2. What do they give?

    • 1, an underestimate
    • 1, an overestimate
    • 4, an underestimate
    • 2, the exact value

  8. Two left-endpoint strips estimate the area under y equals x squared from 0 to 2. What do they give?

    • 1, an underestimate
    • 5, an overestimate
    • 8 over 3, the exact value

  9. Mia integrates x squared minus 1 from 0 to 2, gets 2 over 3, and reports an area of 2 over 3. What did she miss?

    • Nothing. 2 over 3 is the area.
    • The antiderivative, which is wrong.
    • The dip below the axis, which must flip positive.

  10. Evaluate the integral from 0 to 2 of 2x plus 1 dx.

    Answer: ______________

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Answer key

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

From rectangles to exact area W1-mt_S2IO2S7kgd-s1

  1. [x²] from 1 to 3 = 9 - 1 = 8 · The antiderivative is x², and 3² - 1² = 8.
  2. 8 · [x²] from 1 to 3 = 9 - 1 = 8.
  3. Square bracket x squared, from 1 to 3, giving 9 minus 1, which is 8 · The antiderivative of 2x is x squared, and 9 minus 1 is 8.
  4. True · Brackets plus limits collapse to one number. No limits means a family.
  5. False · The signed integral is 2 over 3, but the true area flips the dip positive to 2.
  6. False · Tom is wrong: the signed integral is 2/3 but the true area is 2, because area below the axis must be flipped positive.
  7. 1, an underestimate · Left endpoints give heights 0 and 1 over width 1, so 1, which misses the rising curve and sits below 8/3.
  8. 1, an underestimate · Heights 0 and 1 over width 1 sum to 1, below the exact 8 over 3.
  9. The dip below the axis, which must flip positive. · She kept the signed total. Splitting at the root and flipping gives 2.
  10. 6 · Brackets give x squared plus x, and 6 minus 0 is 6.
Worksheet · LightMySky