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 = 4y = 3y = 11431

Evaluate I = ∫14 ∫13 (3x·ey) dy dx over the rectangle 1 ≤ x ≤ 4, 1 ≤ y ≤ 3.

f = 3x·ey1 ≤ x ≤ 41 ≤ y ≤ 3
I =
2.Easy
xyx = 1x = 4y = 4y = 11441

Evaluate I = ∫14 ∫14 (y² + 3) dy dx over the rectangle 1 ≤ x ≤ 4, 1 ≤ y ≤ 4.

f = y² + 31 ≤ x ≤ 41 ≤ y ≤ 4
I =
3.Medium
xy0.5−1∇fP

For f(x, y) = 3y·exy, find the maximum rate of change of f at the point (0.5, −1), that is |∇f|.

f(x, y) = 3y·exyP = (0.5, −1)
|∇f| =
4.Medium

z = −3x³y³ + x³ + y³ + 2xy with x = −t + 2 and y = t² − 2. Use the chain rule to find dz/dt at t = 0.

z = −3x³y³ + x³ + y³ + 2xyx = −t + 2y = t² − 2t = 0
dz/dt =
5.Hard
xyy = 1.5y = x1.51.5

The integral I = ∫01.5 ∫x1.5 ey² dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = ey²0 ≤ x ≤ y ≤ 1.5
I =
6.Hard
xyy = 2y = x22

The integral I = ∫02 ∫x2 sin(y²) dy dx cannot be done in this order. Reverse the order of integration and evaluate it.

f = sin(y²)0 ≤ x ≤ y ≤ 2
I =

Answer key

  1. 1. I = 390.8
  2. 2. I = 90
  3. 3. |∇f| = 2.034
  4. 4. dz/dt = −296
  5. 5. I = 4.244
  6. 6. I = 0.8268

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