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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
xyx = 1y = 2y = 1121

Evaluate I = ∫01 ∫12 (−x²y + 3) dy dx over the rectangle 0 ≤ x ≤ 1, 1 ≤ y ≤ 2.

f = −x²y + 30 ≤ x ≤ 11 ≤ y ≤ 2
I =
2.Easy
xyx = 2x = 4y = 5y = 22452

Evaluate I = ∫24 ∫25 (2y² + 6) dy dx over the rectangle 2 ≤ x ≤ 4, 2 ≤ y ≤ 5.

f = 2y² + 62 ≤ x ≤ 42 ≤ y ≤ 5
I =
3.Medium

z = −2xy³ − 4x² − 3y² − xy with x = −2t − 1 and y = 2t² − 1. Use the chain rule to find dz/dt at t = −1.

z = −2xy³ − 4x² − 3y² − xyx = −2t − 1y = 2t² − 1t = −1
dz/dt =
4.Medium
xy02024PQ

For f(x, y) = 2 sin(2x + 3y), find the directional derivative Duf at P(1, 1) in the direction towards Q(3, 4).

f(x, y) = 2 sin(2x + 3y)P = (1, 1)Q = (3, 4)
Duf =
5.Hard
xyx = 1y = x1

The integral I = ∫01 ∫y1 ex² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = ex²0 ≤ y ≤ x ≤ 1
I =
6.Hard
xyx = 2y = 3x26

Evaluate I = ∬T (4x − 2y) dA over the triangle T bounded by y = 0, x = 2 and y = 3x.

f = 4x − 2y0 ≤ x ≤ 20 ≤ y ≤ 3x
I =

Answer key

  1. 1. I = 2.5
  2. 2. I = 192
  3. 3. dz/dt = 74
  4. 4. Duf = 2.046
  5. 5. I = 0.8591
  6. 6. I = 8

Full step-by-step solutions: practise this topic on examproblems.com.