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courses › Calculus

Integrals

level 41 course

Accumulation: areas, averages, and the fundamental theorem.

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the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

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The idea

A definite integral, written ∫ₐᵇ f(x) dx, adds up f from x = a to x = b: the area under the graph, with parts below the axis counting as negative.

To work one out, find an antiderivative F, a function whose derivative is f, and take F(b) − F(a). That shortcut is the fundamental theorem of calculus. It also runs the other way: differentiating ∫ₐˣ g(t) dt gives back g(x).

Integrals give averages too. The average value of f over an interval is its integral divided by the interval's length.

Techniques

Antiderivative, then top minus bottom

A definite integral of a polynomial, or an average value.

  1. Raise each power by one and divide by the new power: 6x² becomes 2x³.
  2. A plain number c becomes cx.
  3. Answer: F(top limit) − F(bottom limit).
  4. For an average value, divide that by the interval’s length, b − a.
worked example

Example: What is ∫₁² (3x² + 4x) dx?

  1. Antiderivative: F(x) = x³ + 2x².
  2. F(2) = 8 + 8 = 16. F(1) = 1 + 2 = 3.
  3. 16 − 3 = 13.

Answer: 13

Differentiate an integral

F(x) = ∫ₐˣ g(t) dt, and you need F′ at a number.

  1. F′(x) = g(x): differentiating undoes the integrating.
  2. Put the number into g. The lower limit a doesn’t matter.
worked example

Example: F(x) = ∫₂ˣ (t² − 3t) dt. What is F′(4)?

  1. F′(x) = x² − 3x, the integrand with t replaced by x.
  2. F′(4) = 16 − 12 = 4.

Answer: 4

Area between curves

Two curves meet at x = p and x = q, and you need the area between them.

  1. Test one x between p and q to see which curve is on top.
  2. Subtract: top minus bottom, as one polynomial.
  3. Integrate that from p to q. The area comes out positive.
worked example

Example: The curves y = x² and y = 2x meet at x = 0 and x = 2. What is the area of the region between them? Enter a fraction or a whole number.

  1. At x = 1, 2x is 2 and x² is 1, so y = 2x is on top.
  2. Top minus bottom: 2x − x². Antiderivative: x² − x³/3.
  3. At 2: 4 − 8/3 = 4/3. At 0 it is 0.

Answer: 4/3

Tips by skill

  • TipDefinite integrals: Raise each power by one and divide by the new power. Then take F(top limit) − F(bottom limit).
  • TipAverage value: Integrate f over the interval, then divide by the interval’s length, b − a.
  • TipFundamental theorem: The derivative of ∫ₐˣ g(t) dt is g(x). Put the number into the integrand; the lower limit doesn’t matter.
  • TipArea between curves: Test a point between the crossings to see which curve is on top, then integrate top minus bottom between them.

Watch out for

  • Raising the power without dividing by the new power. The antiderivative of 3x² is x³, not 3x³.
  • Giving the integral as the average value. Divide it by the interval’s length.
  • Working out F(c) when asked for F′(c). The theorem hands you the integrand at c, with nothing to integrate.
  • Subtracting the top curve from the bottom one. An area is never negative.

skills · practice stats

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

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.

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