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 = 1y = 2y = 1121

Evaluate I = ∫01 ∫12 (x²y + 4) dy dx over the rectangle 0 ≤ x ≤ 1, 1 ≤ y ≤ 2.

f = x²y + 40 ≤ x ≤ 11 ≤ y ≤ 2
I =
2.Easy
xyx = 2x = 4y = 5y = 22452

Evaluate I = ∫24 ∫25 (x·e−y) dy dx over the rectangle 2 ≤ x ≤ 4, 2 ≤ y ≤ 5.

f = x·e−y2 ≤ x ≤ 42 ≤ y ≤ 5
I =
3.Medium

z = 3x²y² − 3x² − 4y² + 3xy with x = 2t and y = 2t². Use the chain rule to find dz/dt at t = 1.

z = 3x²y² − 3x² − 4y² + 3xyx = 2ty = 2t²t = 1
dz/dt =
4.Medium
xy−2−2∇fP

For f(x, y) = 2x³y³ − 5x² + 3y³ + 4xy, find the maximum rate of change of f at the point (−2, −2), that is |∇f|.

f(x, y) = 2x³y³ − 5x² + 3y³ + 4xyP = (−2, −2)
|∇f| =
5.Hard
xyx = 0.8y = x0.8

The integral I = ∫00.8 ∫y0.8 1/(1 + x²)² dx dy cannot be done in this order. Reverse the order of integration and evaluate it.

f = 1/(1 + x²)²0 ≤ y ≤ x ≤ 0.8
I =
6.Hard
xyx = 1y = 3x13

Evaluate I = ∬T (4x + y) dA over the triangle T bounded by y = 0, x = 1 and y = 3x.

f = 4x + y0 ≤ x ≤ 10 ≤ y ≤ 3x
I =

Answer key

  1. 1. I = 4.5
  2. 2. I = 0.7716
  3. 3. dz/dt = 236
  4. 4. |∇f| = 243.5
  5. 5. I = 0.1951
  6. 6. I = 5.5

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