courses › Calculus

Accumulation: areas, averages, and the fundamental theorem.

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From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Definite integrals par 30 s not tried yet

  worked example

  What is ∫₂³ (−4x − 4) dx?

  Answer: -14

  1. Antiderivative: F(x) = −2x² − 4x.
  2. F(3) = −30 and F(2) = −16.
  3. So the integral is −30 − (−16) = −14.

• Average value par 35 s not tried yet

  worked example

  What is the average value of f(x) = x² on the interval [2, 5]?

  Answer: 13

  1. Average value = (integral of f over the interval) ÷ (length of the interval).
  2. An antiderivative is F(x) = x³/3, and F(5) − F(2) = 125/3 − 8/3 = 39.
  3. The length is 5 − 2 = 3, so the average value is 39 ÷ 3 = 13.

• Fundamental theorem par 25 s not tried yet

  worked example

  F(x) = ∫₃ˣ (−3t + 5) dt. What is F′(−3)?

  Answer: 14

  1. The fundamental theorem of calculus: the derivative of ∫₃ˣ g(t) dt is g(x), so F′(x) = −3x + 5.
  2. The lower limit 3 doesn’t matter: F′(−3) = −3(−3) + 5 = 9 + 5 = 14.

• Area between curves par 45 s not tried yet

  worked example

  The curves y = x² + 2x − 9 and y = 2x² − 3x − 3 meet at x = 2 and x = 3. What is the area of the region between them? Enter a fraction or a whole number.

  Answer: 1/6

  1. Between x = 2 and x = 3, y = x² + 2x − 9 is on top.
  2. Top minus bottom is −x² + 5x − 6; an antiderivative is −x³/3 + 5x²/2 − 6x.
  3. It is −9/2 at x = 3 and −14/3 at x = 2: −9/2 − (−14/3) = 1/6.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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