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 = 2x = 3y = 4y = 22342

Evaluate I = ∫23 ∫24 (3x·ey) dy dx over the rectangle 2 ≤ x ≤ 3, 2 ≤ y ≤ 4.

f = 3x·ey2 ≤ x ≤ 32 ≤ y ≤ 4
I =
2.Easy
xyx = 2x = 5y = 2y = 12521

Evaluate I = ∫25 ∫12 (x·e0.5y) dy dx over the rectangle 2 ≤ x ≤ 5, 1 ≤ y ≤ 2.

f = x·e0.5y2 ≤ x ≤ 51 ≤ y ≤ 2
I =
3.Medium
xy−10.5P

Use the tangent plane of f(x, y) = 2 ln(x² + y²) at (−1, 0.5) to approximate f(−0.9, 0.3). Give the linear approximation L.

f(x, y) = 2 ln(x² + y²)(a, b) = (−1, 0.5)(x, y) = (−0.9, 0.3)
L =
4.Medium

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

z = sin(x + 2y)x = −3ty = t² + 1t = 0
dz/dt =
5.Hard
xyy = 2y = x22

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

f = cos(y²)0 ≤ x ≤ y ≤ 2
I =
6.Hard
xyy = xy = x²11

Evaluate I = ∬R x dA, where R is the region between the parabola y = x² and the line y = x.

y = x²y = xf = x
I =

Answer key

  1. 1. I = 354.1
  2. 2. I = 22.46
  3. 3. L = −0.1937
  4. 4. dz/dt = 1.248
  5. 5. I = −0.3784
  6. 6. I = 0.08333

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