Multivariable Calculus Review Problems

Harini ChandramouliMATH2374

chand409@umn.edu

Problem 1. This semester we have discussed many different types of integrals: doubleintegrals, triple integrals, path integrals, line integrals, scalar surface integrals, and surfaceintegrals of a vector field. Below, write out the definitions of each of the integrals listedabove. When do you use each type of integral? What do they physically represent? How dothey relate to each other (think of some theorems we’ve learned)?

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Problem 2. Approximate the value√

8 cos(0.08) + e−0.06.

Problem 3. Let z = x2 sin(y), where x = s2 + t2 and y = 2st. Find ∂z∂s

and ∂z∂t

.

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Problem 4. Find the extreme values of f(x, y) = e−xy within the region x2 + 4y2 ≤ 1.

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Problem 5. Evaluate each of the following integrals.

(a) ∫∫D

yex dA

where D is the triangular region with vertices (0, 0), (2, 4), and (6, 0).

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(b)

2∫−2

√4−y2∫

−√

4−y2

2∫√

x2+y2

z dz dx dy

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Problem 6. Find the volume of the solid that lies above the cone z =√x2 + y2 and below

the sphere x2 + y2 + z2 = 5z.

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Problem 7. Evaluate ∫C

y2 dx+ x dy

where C is the line segment from (−2, 4) to (4, 6).

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Problem 8. Evaluate ∫C

F · ds

where F (x, y) = (4x3y2 − 2xy3, 2x4y − 3x2y2 + 4y3) where C is parametrized by x = t +sin(πt) and y = 2t + cos(πt) for 0 ≤ t ≤ 1. (Hint: It might be helpful to check if the givenvector field is conservative first.)

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Problem 9. Find the area of the surface with vector equation r(u, v) = (u cos(v), u sin(v), v)with 0 ≤ u ≤ 1 and 0 ≤ v ≤ π.

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Problem 10. Evaluate the line integral∫C

(y + sin(x)) dx+ (z2 + cos(y)) dy + x3 dz

where C is the curve parametrized by r(t) = (sin(t), cos(t), sin(2t)) for 0 ≤ t ≤ 2π. Observethat C lies on the surface z = 2xy.

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Problem 11. Calculate the flux of F across S if F (x, y, z) = (x2z3, 2xyz3, xz4) and S isthe surface of the box with vertices (±1,±2,±3).

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Problem 12. Find the points on the elliptic paraboloid z = 4x2 + y2 closest to (0, 0, 8).

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Problem 13. Consider two spheres of radius one that intersect such that the leftmost pointof the right hand sphere is at the center of the left hand sphere. Find the volume of theregion of intersection. (Hint: It might help to try the easier problem first of considering twointersecting circles of radius 1, and then find the area of intersection.)

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Problem 14. Using triple integrals, prove that the volume of a parallelpiped is equal tothe scalar triple product.

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Problem 15. Let S be the box formed by the planes x = 0, y = 0, z = 0, x = 1, y = 1, andz = 1, excluding the top face, oriented outward. Let F = (yz2, xy, x2). Evaluate∫∫

S

F dS.

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Problem 16. Find the volume of the solid bounded by (both part of) the cone z2 = x2 +y2

and the sphere x2 + y2 + z2 = 1. Try this using both double and triple integrals.

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Problem 17. Let D be the region outside of D1 ={

(x, y)∣∣x2 + (y − 2)2 ≤ 4

}but inside

the region D2 ={

(x, y)∣∣x2 + (y − 3)2 ≤ 9

}. Evaluate∫∫

D

(x+ y) dA.

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Problem 18. Find the minimum distance between the lines (x, y, z) = t (1, 1, 1) and (x, y, z) =(−1, 2, 1) + s (1, 0,−1) for t, s ∈ R.

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Problem 19. Find the volume of the piece of the sphere x2 + y2 + z2 = 1 cut off by theplane x+ y + z = 1.

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Problem 20. Let S be the lateral surface of a cone of radius 1 and height 2 parametrizedas (x, y, z) = (r cos(θ), 4 sin(θ), 2(1− r)) oriented inward. Let F = (x, y, 0). Evaluate∫∫

S

F dS.

(Hint: The formula for the volume of a cone is V = 13πr2h.)

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Problem 21. This is a spatial reasoning problem that is meant to help with your visual-ization of triple integrals. Describe (verbally, pictorally, spiritually, whatever you want) theregion of intersection of three cylinders of radius 1 if the cylinders all intersect orthogonally.

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Note: These questions were put together by Justin Gaffur and Yara Skaf from the Universityof Pittsburgh. Some problems were taken from Marsden and Tromba’s Vector Calculus,Sixth Edition textbook.

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