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

Evaluate I = ∫₁⁴ ∫₂³ (2x·e^−y) dy dx over the rectangle 1 ≤ x ≤ 4, 2 ≤ y ≤ 3.

f = 2x·e^−y1 ≤ x ≤ 42 ≤ y ≤ 3

I =

2.Easy

Evaluate I = ∫₁³ ∫₂⁴ (4xy + 2) dy dx over the rectangle 1 ≤ x ≤ 3, 2 ≤ y ≤ 4.

f = 4xy + 21 ≤ x ≤ 32 ≤ y ≤ 4

I =

3.Medium

The tangent plane to z = 3 ln(x² + y²) at the point where (x, y) = (0.5, 0.5) can be written as z = px + qy + c. Find c.

f(x, y) = 3 ln(x² + y²)(a, b) = (0.5, 0.5)

c =

4.Medium

z = 5xy² + 4x³ − 5y² + xy with x = −t and y = 2t² − 2. Use the chain rule to find dz/dt at t = 1.

z = 5xy² + 4x³ − 5y² + xyx = −ty = 2t² − 2t = 1

dz/dt =

5.Hard

Use polar coordinates to evaluate I = ∬_R (x² + y²) dA, where R is the ring 1 ≤ x² + y² ≤ 9.

f = x² + y²1 ≤ r ≤ 30 ≤ θ ≤ 2π

I =

6.Hard

Use polar coordinates to evaluate I = ∬_R (x² + y²) dA, where R is the disc x² + y² ≤ 9.

f = x² + y²0 ≤ r ≤ 30 ≤ θ ≤ 2π

I =

Answer key

1. I = 1.283
2. I = 104
3. c = −8.079
4. dz/dt = −16
5. I = 125.7
6. I = 127.2

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