The document contains a test with 9 problems involving calculus concepts like double and triple integrals in Cartesian, polar, and spherical coordinates, line integrals, vector fields, and surface integrals. Students are instructed to show all work algebraically and given reference sheets on derivatives and anti-derivatives. The test covers multiple topics within multivariate calculus.

1. MAT225 TEST4A Name:
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(1) Double Integrals (Cartesian Coordinates)
0
1
∫
𝑦
2
1
2
∫ 𝑒
−𝑥
2
𝑑𝑥𝑑𝑦
(1a) Describe the region R over which we are integrating.
(1b) Rewrite the given integral such that the area element 𝑑𝐴 = 𝑑𝑦𝑑𝑥.
(1b) Evaluate the new integral over said region R.
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2. MAT225 TEST4A Name:
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3. MAT225 TEST4A Name:
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(2) Double Integrals (Polar Coordinates)
0
3
∫
0
9−𝑥
2
∫ 𝑥 𝑑𝑦𝑑𝑥
(2a) Describe the region R over which we are integrating.
(2b) Rewrite the given integral such that the area element 𝑑𝐴 = 𝑟𝑑𝑟𝑑θ.
(2b) Evaluate the new integral over said region R.
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4. MAT225 TEST4A Name:
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TEST4A page: 4

5. MAT225 TEST4A Name:
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(3) Double Integrals (Polar Coordinates)
𝑆 = ∫
𝑅
∫ 1 + 𝑓𝑥
2
+ 𝑓𝑦
2
𝑑𝐴
(3) Find the surface area of in the first octant.
𝑧 = 16 − 𝑥
2
− 𝑦
2
(3a) Write the integral such that the area element 𝑑𝐴 = 𝑑𝑦𝑑𝑥.
(3b) Rewrite this integral such that the area element 𝑑𝐴 = 𝑟𝑑𝑟𝑑θ.
(3c) Evaluate the new integral to find S.
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6. MAT225 TEST4A Name:
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7. MAT225 TEST4A Name:
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(4) Triple Integrals (Cylindrical Coordinates)
(4) Find the volume of the solid bounded by and z=0.
𝑧 = 16 − 𝑥
2
− 𝑦
2
(4a) Write the integral such that the volume element 𝑑𝑉 = 𝑑𝑧𝑑𝑦𝑑𝑥.
(4b) Rewrite this integral such that the volume element 𝑑𝑉 = 𝑟𝑑𝑧𝑑𝑟𝑑θ.
(4c) Evaluate the new integral to find V.
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8. MAT225 TEST4A Name:
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TEST4A page: 8

9. MAT225 TEST4A Name:
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(5) Triple Integrals (Spherical Coordinates)
𝑥 = 𝑟 𝑐𝑜𝑠(θ) = ρ 𝑠𝑖𝑛(ϕ) 𝑐𝑜𝑠(θ)
𝑦 = 𝑟 𝑠𝑖𝑛(θ) = ρ 𝑠𝑖𝑛(ϕ) 𝑠𝑖𝑛(θ)
𝑧 = ρ 𝑐𝑜𝑠(ϕ)
ρ
2
= 𝑟
2
+ 𝑧
2
= 𝑥
2
+ 𝑦
2
+ 𝑧
2
(5) Find the coordinates of the center of mass ( ) for the solid bounded by the
𝑥, 𝑦, 𝑧
upper half of the sphere and z=0 with variable density
ρ = 6 δ(ρ, ϕ, θ) = 1 +
ρ
4
(5a) Find the total mass, m=∫∫
𝑄
∫ δ(ρ, ϕ, θ) 𝑑𝑉
(5b) Find 𝑥 =
1
𝑚
∫∫
𝑄
∫ 𝑥 δ(ρ, ϕ, θ) 𝑑𝑉
(5c) Find 𝑦 =
1
𝑚
∫∫
𝑄
∫ 𝑦 δ(ρ, ϕ, θ) 𝑑𝑉
TEST4A page: 9

10. MAT225 TEST4A Name:
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TEST4A page: 10

11. MAT225 TEST4A Name:
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(5) Triple Integrals (Spherical Coordinates)
𝑥 = 𝑟 𝑐𝑜𝑠(θ) = ρ 𝑠𝑖𝑛(ϕ) 𝑐𝑜𝑠(θ)
𝑦 = 𝑟 𝑠𝑖𝑛(θ) = ρ 𝑠𝑖𝑛(ϕ) 𝑠𝑖𝑛(θ)
𝑧 = ρ 𝑐𝑜𝑠(ϕ)
ρ
2
= 𝑟
2
+ 𝑧
2
= 𝑥
2
+ 𝑦
2
+ 𝑧
2
(5) Find the coordinates of the center of mass ( ) for the solid bounded by the
𝑥, 𝑦, 𝑧
upper half of the sphere and z=0 with variable density
ρ = 6 δ(ρ, ϕ, θ) = 1 +
ρ
4
(5d) Find 𝑧 =
1
𝑚
∫∫
𝑄
∫ 𝑧 δ(ρ, ϕ, θ) 𝑑𝑉
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12. MAT225 TEST4A Name:
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TEST4A page: 12

13. MAT225 TEST4A Name:
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(6) Line Integrals
Find the work done by the force field F(x,y) = <x,2y> done on a particle
moving along the path C: y = from the point (0,0) to the point (2,8).
𝑥
3
F = <M,N>
<M,N><dx,dy> =
𝑊 =
𝐶
∫ 𝐹𝑑𝑟 =
𝐶
∫
𝐶
∫ 𝑀𝑑𝑥 + 𝑁𝑑𝑦
(6a) Parametrize the path C in terms of a single parameter t.
(6b) Write the Line Integral for Work in terms of t.
(6c) Evaluate your integral from t = 0 to t = 2.
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14. MAT225 TEST4A Name:
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15. MAT225 TEST4A Name:
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(7) Fundamental Theorem of Line Integrals
F = <M,N> = <2xy, 𝑥
2
+ 𝑦
2
>
(7a) Show that F is a Conservative Vector Field.
(7b) Find the Potential Function f(x,y) for the Vector Field F.
(7c) Evaluate W = using f(x,y) from (5,0) to (0,4) over the path C:
𝐶
∫ 𝑀𝑑𝑥 + 𝑁𝑑𝑦
𝑥
2
25
+
𝑦
2
16
= 1
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16. MAT225 TEST4A Name:
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TEST4A page: 16

17. MAT225 TEST4A Name:
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(8) Green’s Theorem for Work in the Plane
𝐹(𝑥, 𝑦) =< 𝑀, 𝑁 >=< 𝑦
2
, 𝑥
2
>
C: CCW once about 𝑦 = 𝑥
2
𝑎𝑛𝑑 𝑦 = 𝑥
(8a) Parametrize the path C1: along the curve from (0,0) to (1,1) in terms of t.
𝑦 = 𝑥
2
(8b) Use this parametrization to find the work done:
<M,N><dx,dy> =
𝑊 =
𝐶1
∫
𝐶1
∫ 𝑀𝑑𝑥 + 𝑁𝑑𝑦
TEST4A page: 17

18. MAT225 TEST4A Name:
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TEST4A page: 18

19. MAT225 TEST4A Name:
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(8) Green’s Theorem for Work in the Plane
𝐹(𝑥, 𝑦) =< 𝑀, 𝑁 >=< 𝑦
2
, 𝑥
2
>
C: CCW once about 𝑦 = 𝑥
2
𝑎𝑛𝑑 𝑦 = 𝑥
(8c) Parametrize the path C2: along the curve from (1,1) to (0,0) in terms of t.
𝑦 = 𝑥
(8d) Use this parametrization to find the work done:
<M,N><dx,dy> =
𝑊 =
𝐶2
∫
𝐶2
∫ 𝑀𝑑𝑥 + 𝑁𝑑𝑦
TEST4A page: 19

20. MAT225 TEST4A Name:
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TEST4A page: 20

21. MAT225 TEST4A Name:
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(8) Green’s Theorem for Work in the Plane
𝐹(𝑥, 𝑦) =< 𝑀, 𝑁 >=< 𝑦
2
, 𝑥
2
>
C: CCW once about 𝑦 = 𝑥
2
𝑎𝑛𝑑 𝑦 = 𝑥
(8e) Verify Green’s Theorem for Work in the Plane.
TEST4A page: 21

22. MAT225 TEST4A Name:
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TEST4A page: 22

23. MAT225 TEST4A Name:
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(9) Surface Integrals
Given the density function ρ
ρ(𝑥, 𝑦, 𝑧) = 𝑥 − 2𝑦 + 𝑧
find the mass of the planar region S
𝑧 = 4 − 𝑥, 0 ≤ 𝑥 ≤ 4, 0 ≤ 𝑦 ≤ 3
(9a) State the surface area element such that dA = dydx.
𝑑𝑆 = 1 + 𝑓𝑥
2
+ 𝑓𝑦
2
𝑑𝐴
(9b) Evaluate the surface integral
𝑆 = ∫
𝑅
∫ ρ(𝑥, 𝑦, 𝑧)𝑑𝑆
TEST4A page: 23

24. MAT225 TEST4A Name:
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TEST4A page: 24

25. MAT225 TEST4A Name:
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Reference Sheet: Derivatives You Should Know Cold!
Power Functions:
𝑑
𝑑𝑥
𝑥
𝑛
= 𝑛𝑥
𝑛−1
Trig Functions:
𝑑
𝑑𝑥
𝑠𝑖𝑛(𝑥) = 𝑐𝑜𝑠(𝑥)
𝑑
𝑑𝑥
𝑐𝑜𝑠(𝑥) = − 𝑠𝑖𝑛(𝑥)
𝑑
𝑑𝑥
𝑡𝑎𝑛(𝑥) = 𝑠𝑒𝑐
2
(𝑥)
𝑑
𝑑𝑥
𝑐𝑜𝑡(𝑥) = − 𝑐𝑠𝑐
2
(𝑥)
𝑑
𝑑𝑥
𝑠𝑒𝑐(𝑥) = 𝑠𝑒𝑐(𝑥) 𝑡𝑎𝑛(𝑥)
𝑑
𝑑𝑥
𝑐𝑠𝑐(𝑥) = − 𝑐𝑠𝑐(𝑥) 𝑐𝑜𝑡(𝑥)
Transcendental Functions:
𝑑
𝑑𝑥
𝑒
𝑥
= 𝑒
𝑥 𝑑
𝑑𝑥
𝑎
𝑥
= 𝑙𝑛(𝑎) 𝑎
𝑥
𝑑
𝑑𝑥
𝑙𝑛(𝑥) =
1
𝑥
𝑑
𝑑𝑥
𝑙𝑜𝑔𝑎
(𝑥) =
1
𝑙𝑛(𝑎)
1
𝑥
Inverse Trig Functions:
𝑑
𝑑𝑥
𝑠𝑖𝑛
−1
(𝑥) =
1
1−𝑥
2
𝑑
𝑑𝑥
𝑐𝑜𝑠
−1
(𝑥) =
−1
1−𝑥
2
𝑑
𝑑𝑥
𝑡𝑎𝑛
−1
(𝑥) =
1
1+𝑥
2
𝑑
𝑑𝑥
𝑐𝑜𝑡
−1
(𝑥) =
−1
1+𝑥
2
Product Rule:
𝑑
𝑑𝑥
𝑓(𝑥) 𝑔(𝑥) = 𝑓(𝑥) 𝑔'(𝑥) + 𝑔(𝑥) 𝑓'(𝑥)
Quotient Rule:
𝑑
𝑑𝑥
𝑓(𝑥)
𝑔(𝑥)
=
𝑔(𝑥) 𝑓'(𝑥) − 𝑓(𝑥) 𝑔'(𝑥)
𝑔
2
(𝑥)
Chain Rule:
𝑑
𝑑𝑥
𝑓(𝑔(𝑥)) = 𝑓'(𝑔(𝑥)) 𝑔'(𝑥)
Difference Quotient:
f’(x) =
ℎ 0
lim
→
𝑓(𝑥+ℎ) − 𝑓(𝑥)
ℎ
TEST4A page: 25

26. MAT225 TEST4A Name:
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Reference Sheet: Anti-Derivatives You Should Know Cold!
Power Functions:
∫ 𝑥
𝑛
𝑑𝑥 = 𝑛𝑥
𝑛−1
Trig Functions:
∫ 𝑐𝑜𝑠(𝑥)𝑑𝑥 = 𝑠𝑖𝑛(𝑥) + 𝐶 ∫ 𝑠𝑖𝑛(𝑥)𝑑𝑥 = − 𝑐𝑜𝑠(𝑥) + 𝐶
∫ 𝑠𝑒𝑐
2
(𝑥)𝑑𝑥 = 𝑡𝑎𝑛(𝑥) + 𝐶 ∫ 𝑐𝑠𝑐
2
(𝑥)𝑑𝑥 = − 𝑐𝑜𝑡(𝑥) + 𝐶
∫ 𝑠𝑒𝑐(𝑥)𝑡𝑎𝑛(𝑥)𝑑𝑥 = 𝑠𝑒𝑐(𝑥) + 𝐶 ∫ 𝑐𝑠𝑐(𝑥)𝑐𝑜𝑠(𝑥)𝑑𝑥 = − 𝑐𝑠𝑐(𝑥) + 𝐶
Transcendental Functions:
∫ 𝑒
𝑥
𝑑𝑥 = 𝑒
𝑥
+ 𝐶 ∫ 𝑎
𝑥
𝑑𝑥 =
𝑎
𝑥
𝑙𝑛(𝑎)
+ 𝐶
∫
1
𝑥
𝑑𝑥 = 𝑙𝑛(𝑥) + 𝐶 ∫
1
𝑙𝑛(𝑎)
1
𝑥
𝑑𝑥 = 𝑙𝑜𝑔𝑎
(𝑥) + 𝐶
Inverse Trig Functions:
∫
1
1−𝑥
2
𝑑𝑥 = 𝑠𝑖𝑛
−1
(𝑥) + 𝐶 ∫
−1
1−𝑥
2
𝑑𝑥 = 𝑐𝑜𝑠(𝑥) + 𝐶
∫
1
1+𝑥
2 𝑑𝑥 = 𝑡𝑎𝑛
−1
(𝑥) + 𝐶 ∫
−1
1+𝑥
2 𝑑𝑥 = 𝑐𝑜𝑡
−1
(𝑥) + 𝐶
Integration By Parts (Product Rule):
∫ 𝑢𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢 + 𝐶
Integration By Partial Fractions Example (Quotient Rule):
∫
𝑑𝑥
𝑥(𝑥+1)
= ∫
𝐴𝑑𝑥
𝑥
+ ∫
𝐵𝑑𝑥
𝑥+1
TEST4A page: 26