Double Integrals over General Regions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Integrating between two curves

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

  1. Between y = x and y = x squared with 0 <= x <= 1, with x outside, how should the inner integral run?

    • inner from x to x squared
    • inner from 0 to x
    • inner from x squared to x

  2. Sam says the volume under the flat surface z = 2 over the unit square is 2. Is Sam right?

    Circle one:   True   False

  3. The region R is the rectangle 0 <= x <= 2, 0 <= y <= 3. What is the double integral of 1 over R?

    Answer: ______________

  4. Sam says the volume under the flat surface z = 2 over the unit square is 2. Is Sam right?

    Circle one:   True   False

  5. What is the integral from x = 0 to 1 of the inner integral from y = 0 to 1 of (x + y)?

    Answer: ______________

  6. Consider the region between y = x and y = x squared with 0 <= x <= 1. Which iterated integral gives its area?

    • The integral from 0 to 1 of the inner integral from x squared to x dy dx
    • The integral from 0 to 1 of the inner integral from x to x squared dy dx
    • The integral from 0 to 1 of the inner integral from 0 to x dy dx
    • The integral from 0 to 1 of the inner integral from x squared to 1 dy dx

  7. The region R is the rectangle 0 <= x <= 2, 0 <= y <= 3. What is the double integral of 1 over R?

    Answer: ______________

  8. The integral with x outside 0 to 1 and y from x to 1 covers a triangle. Which form reverses the order?

    • y outside 0 to 1, x from 0 to y
    • y outside 0 to 1, x from y to 1
    • x outside 0 to 1, y from 0 to x

  9. A region needs two different top curves in dydx order. What should you try?

    • integrate in the same order anyway
    • split into pieces or swap to dxdy
    • change the integrand to 1

  10. Ted writes the area between y = x squared and y = x on [0, 1] with y running from x down to x squared. Is Ted right?

    Circle one:   True   False

LightMySky · lightmysky.comW1-mt_UKS9_AhAWw-s1

Answer key

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

Integrating between two curves W1-mt_UKS9_AhAWw-s1

  1. inner from x squared to x · On [0, 1] the line y = x sits above the parabola, so y runs from x squared up to x.
  2. True · Sam is right: volume is base area times height, 1 times 2 = 2.
  3. 6 · Integrating 1 over a region gives its area, and this rectangle has area 2 times 3 = 6.
  4. True · Sam is right: volume is base area times height, 1 times 2 = 2.
  5. 1 · Split the sum: the x part gives 1/2 and the y part gives 1/2, so the total is 1.
  6. The integral from 0 to 1 of the inner integral from x squared to x dy dx · On [0, 1] the line y = x sits above the parabola, so y runs from x squared up to x.
  7. 6 · Integrating 1 over a region gives its area, and this rectangle has area 2 times 3 = 6.
  8. y outside 0 to 1, x from 0 to y · The region is 0 <= x <= y <= 1, so with y outside, x runs from 0 to y.
  9. split into pieces or swap to dxdy · A fighting order means the slices cut the region badly; swapping usually fixes it.
  10. False · The lower curve goes first: x squared is below x there.
Worksheet · LightMySky