Multivariable calculus worksheet with answers

Free printable Multivariable calculus worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
xyx = 1x = 3y = 2y = 11321

Evaluate I = ∫13 ∫12 (3y − 2) dy dx over the rectangle 1 ≤ x ≤ 3, 1 ≤ y ≤ 2.

f = 3y − 21 ≤ x ≤ 31 ≤ y ≤ 2
I =
2.Easy
xyx = 2y = 3y = 2232

Evaluate I = ∫02 ∫23 (−3xy²) dy dx over the rectangle 0 ≤ x ≤ 2, 2 ≤ y ≤ 3.

f = −3xy²0 ≤ x ≤ 22 ≤ y ≤ 3
I =
3.Medium
xy024−2024PQ

For f(x, y) = sin(−3x − 3y), find the directional derivative Duf at P(2, 2) in the direction towards Q(3, −2).

f(x, y) = sin(−3x − 3y)P = (2, 2)Q = (3, −2)
Duf =
4.Medium

z = sin(3x + y) with x = 2t and y = −t² − 1. Use the chain rule to find dz/dt at t = 0.

z = sin(3x + y)x = 2ty = −t² − 1t = 0
dz/dt =
5.Hard
xy1.5

Use polar coordinates to evaluate I = ∬R e−(x² + y²) dA, where R is the upper half of the disc x² + y² ≤ 2.25 (y ≥ 0).

f = e−(x² + y²)0 ≤ r ≤ 1.50 ≤ θ ≤ π
I =
6.Hard
xy2

Use polar coordinates to evaluate I = ∬R e−(x² + y²) dA, where R is the disc x² + y² ≤ 4.

f = e−(x² + y²)0 ≤ r ≤ 20 ≤ θ ≤ 2π
I =

Answer key

  1. 1. I = 5
  2. 2. I = −38
  3. 3. Duf = 1.842
  4. 4. dz/dt = 3.242
  5. 5. I = 1.405
  6. 6. I = 3.084

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