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.
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From rectangles to exact area

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

  1. You use F(x) = x² + 7 to evaluate the integral of 2x from 1 to 3. Which endpoint subtraction is correct?

    • (9 + 7) - (1 + 7) = 8
    • (9 + 7) - 1 = 15
    • 9 - (1 + 7) = 1
    • (9 + 7) + (1 + 7) = 24

  2. F(x) = x³ - 2x is an antiderivative of f. What is the integral of f from 2 to 3?

    Answer: ______________

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

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

  4. When it exists, 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. The integral of x - 1 from 0 to 2 is zero, so x - 1 must be zero throughout [0, 2].

    Circle one:   True   False

  7. Use two equal-width left-endpoint rectangles for y = 5 - x on [0, 4]. What is the estimate and its direction of error?

    • 16, an underestimate
    • 8, an overestimate
    • 16, an overestimate
    • 12, an exact estimate

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

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

  9. Mia integrates y = x - 3 on [0, 4], gets -4, and reports geometric area 4 by taking the absolute value of that total. What did she miss?

    • The integral is already positive, so taking an absolute value changes its sign.
    • The antiderivative is x² - 3x, so the endpoint subtraction should give 4.
    • Opposite-signed regions cancel before that final absolute value is taken.

  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. (9 + 7) - (1 + 7) = 8 · The same constant occurs at both endpoints and cancels: 16 - 8 = 8.
  2. 17 · Use F(3) - F(2) = (27 - 6) - (8 - 4) = 21 - 4 = 17.
  3. [x²] from 1 to 3 = 9 - 1 = 8 · The antiderivative of 2x is x squared, and 9 minus 1 is 8.
  4. True · Evaluating the antiderivative at the bracketed limits gives one number. Without limits, the indefinite integral names a family.
  5. False · The signed integral is 2/3. The below-axis part contributes -2/3, so the geometric area is 2/3 + 4/3 = 2.
  6. False · The negative and positive contributions cancel. The function is -1 at x = 0 and 1 at x = 2.
  7. 16, an overestimate · Width 2 and left heights 5 and 3 give 2(5 + 3) = 16. The curve decreases, so this overestimates.
  8. 1, an underestimate · Rectangles with heights 0 and 1 and width 1 give a total of 1, below the exact value 8 over 3.
  9. Opposite-signed regions cancel before that final absolute value is taken. · The separate integrals are -9/2 and 1/2. Their signed total is -4, but geometric area is 9/2 + 1/2 = 5.
  10. 6 · Brackets give x squared plus x, and 6 minus 0 is 6.
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