Improper Integrals and Their Convergence · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Integrate past infinity with limits

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. Which is the correct way to rewrite the integral from 1 to infinity of f(x) dx?

    • Limit as b approaches infinity of the integral from 1 to b of f(x) dx
    • The integral from 1 to a large number like a million
    • f evaluated at infinity minus f evaluated at 1
    • The integral from 1 to 0 with a minus sign

  2. Which is the correct way to rewrite the integral from 1 to infinity of f(x) dx?

    • Plug infinity in for x
    • Take the limit as t grows of the integral from 1 to t
    • Split the integral at 0

  3. For which p does the integral from 1 to infinity of 1/x^p dx converge?

    • p < 1
    • p = 1
    • p > 1

  4. Evaluate the integral from 1 to infinity of 1/x^2 dx.

    Answer: ______________

  5. Evaluate the integral from 0 to 1 of 1/sqrt(x) dx.

    Answer: ______________

  6. The integrand 1 over sqrt x blows up at 0. How do you handle the integral from 0 to 1 of 1 over sqrt x dx?

    • Split it as the limit as a approaches 0 from above of the integral from a to 1
    • Integrate from 0 to 1 directly, ignoring the blow up
    • Split it at x = 1 instead and take two limits
    • Declare it divergent because 0 is in the interval

  7. For which value of p does the integral from 1 to infinity of 1 over x to the p dx converge?

    • p = 2
    • p = 1
    • p = 1/2
    • p = 0

  8. Evaluate the integral from 1 to infinity of 1/x^3 dx.

    Answer: ______________

  9. A friend says the area under 1/x from 1 onward is finite because the curve gets tiny. What is wrong?

    • The curve never touches zero, so the area is zero
    • ln t grows without bound, so the limit diverges
    • Limits never apply to areas

  10. A two-sided infinite integral converges when both halves converge.

    Circle one:   True   False

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Answer key

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Integrate past infinity with limits W1-mt_gvtZPFRA6Z-s1

  1. Limit as b approaches infinity of the integral from 1 to b of f(x) dx · An infinite bound is handled by integrating to a finite b and taking the limit as b grows without bound.
  2. Take the limit as t grows of the integral from 1 to t · Integrate to a finite t first, then let t grow without bound.
  3. p > 1 · The p rule says 1/x^p on [1, infinity) converges exactly when p beats 1.
  4. 1 · The antiderivative 1 - 1/t settles at 1 as t grows.
  5. 2 · The antiderivative 2 sqrt(x) from a small a to 1 approaches 2.
  6. Split it as the limit as a approaches 0 from above of the integral from a to 1 · An interior or endpoint blow up forces a split: stop short at a, integrate, then let a approach the bad point.
  7. p = 2 · The p rule says this integral converges exactly when p exceeds 1, and only p equals 2 clears that bar.
  8. 0.5 · Here p = 3 beats 1, and the limit settles at 1/2.
  9. ln t grows without bound, so the limit diverges · Thin is not enough: ln t still grows past every bound, so the area diverges.
  10. True · Split at a point like 0; the whole converges only if each half does.
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