This document contains a 10-question math exam covering various calculus topics including: 1) Finding roots of polynomials, applying Euler's identity, and using De Moivre's theorem. 2) Evaluating double integrals over described regions using different coordinate systems. 3) Finding surface areas using double integrals and parametrizing surfaces. 4) Finding volumes using triple integrals. 5) Evaluating line integrals of vector fields along described paths. 6) Applying the Fundamental Theorem of Line Integrals to conservative vector fields. 7) Using Green's Theorem to evaluate line integrals of vector fields around paths. 8) Finding mass using surface integrals over a described region with a given density function.

1. MAT225 TEST4A Name:
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RVM (Question 1) Application of Euler’s Identity
(1) Find all three roots:​ x3
+ 8 = 0
Applying De Moivre's Theorem.
Represent a complex number ​z​ as a vector on the Complex Plane
If
z = rcis( ) (cos(θ) sin(θ)) cos(θ) isin(θ)θ = r + i = r + r
Then
cis(nθ)zn
= rn
nd z cis( )a n
1
= √
n
r n
θ+2πk
n, )εZ, n , 0⋁ ( k ≥ 2 ≤ k < n
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(1) Double Integrals
dxdy∫
1
0
∫
2
1
2
y
e−x2
(1a) Describe the region R over which we are integrating.
(1b) Rewrite the given integral such that the area element A ydx.d = d
(1b) Evaluate the new integral over said region R.
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(2) Double Integrals
dydx∫
3
0
∫
√9−x2
0
x
(2a) Describe the region R over which we are integrating.
(2b) Rewrite the given integral such that the area element A drdθ.d = r
(2b) Evaluate the new integral over said region R.
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(3) Double Integrals
dAS = ∫ ∫√1 x y+ f 2 + f 2
(3) Find the surface area of​ in the first octant.6z = 1 − x2
− y2
(3a) Write the integral such that the area element A ydx.d = d
(3b) Rewrite this integral such that the area element A drdθ.d = r
(3c) Evaluate the new integral to find S.
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(4) Triple Integrals
Find the volume of the solid bounded by​ and​6z = 1 − x2
− y2
.z = 0
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(5) Line Integrals
Find the work done by the force field ​F​(x,y) = <x,2y> done on a particle
moving along the path C: x = t, y = from the point (0,0) to the point (2,8).t3
F​ = <M,N>
<M,N><dx,dy> =drW = ∫
C
F = ∫
C
dx dy∫
C
M + N
(5a) Parametrize the path C in terms of a single parameter t.
(5b) Write the Line Integral for Work in terms of t.
(5c) Evaluate your integral from t = 0 to t = 2.
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(6) Fundamental Theorem of Line Integrals
​F​ = <M,N> = <2xy, x2
+ y2
>
(6a) Show that ​F​ is a Conservative Vector Field.
(6b) Find the Potential Function f(x,y) for the Vector Field ​F​.
(6c) Evaluate W = using f(x,y) from (5,0) to (0,4) over the path C:dx dy∫
C
M + N
x2
25 +
y2
16 = 1
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(7) Green’s Theorem for Work in the Plane
(x, ) < , =< ,F y = M N > y2
x2
>
C: CCW once about and yy = x2
= x
(7a) Parametrize the path C​1​: ​ ​along the curve from (0,0) to (1,1) in terms of t.y = x2
(7b) Use this parametrization to find the work done:
<M,N><dx,dy> =W = ∫
C1
dx dy∫
C1
M + N
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(7) Green’s Theorem for Work in the Plane
(x, ) < , =< ,F y = M N > y2
x2
>
C: CCW once about and yy = x2
= x
(7c) Parametrize the path C​2​: ​ ​along the curve from (1,1) to (0,0) in terms of t.y = x
(7d) Use this parametrization to find the work done:
<M,N><dx,dy> =W = ∫
C2
dx dy∫
C2
M + N
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(7) Green’s Theorem for Work in the Plane
(x, ) < , =< ,F y = M N > y2
x2
>
C: CCW once about and yy = x2
= x
(7e) Verify Green’s Theorem for Work in the Plane.
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(8) Surface Integrals
Given the density function ρ
(x, , ) yρ y z = x − 2 + z
find the mass of the planar region S
, 0 , 0z = 4 − x ≤ x ≤ 4 ≤ y ≤ 3
(8a) State the surface area element such that dA = dydx.S dAd = √1 x y+ f 2 + f 2
(8b) Evaluate the surface integral
(x, , )dSS = ∫ ∫
R
ρ y z
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Reference Sheet: Derivatives You Should Know Cold!
Power Functions:
x nxd
dx
n
= n−1
Trig Functions:
sin(x) os(x)d
dx = c cos(x) in(x)d
dx = − s
tan(x) (x)d
dx = sec2
cot(x) (x)d
dx = − csc2
sec(x) ec(x) tan(x)d
dx = s csc(x) sc(x) cot(x)d
dx = − c
Transcendental Functions:
ed
dx
x
= ex
a n(a) ad
dx
x
= l x
ln(x)d
dx = x
1
log (x)d
dx a = 1
ln(a) x
1
Inverse Trig Functions:
sin (x)d
dx
−1
= 1
√1−x2
cos (x)d
dx
−1
= −1
√1−x2
tan (x)d
dx
−1
= 1
1+x2 cot (x)d
dx
−1
= −1
1+x2
Product Rule:
f(x) g(x) (x) g (x) (x) f (x)d
dx = f ′ + g ′
Quotient Rule:
d
dx
f(x)
g(x) = g (x)2
g(x) f (x) − f(x) g (x)′ ′
Chain Rule:
f(g(x)) (g(x)) g (x)d
dx = f′ ′
Difference Quotient:
f’(x) =​ lim
h→0
h
f(x+h) − f(x)
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Reference Sheet: Anti-Derivatives You Should Know Cold!
Power Functions:
dx x∫xn
= n n−1
Trig Functions:
os(x)dx in(x)∫c = s + C in(x)dx os(x)∫s = − c + C
ec (x)dx an(x)∫s 2
= t + C sc (x)dx ot(x)∫c 2
= − c + C
ec(x)tan(x)dx ec(x)∫s = s + C sc(x)cos(x)dx sc(x)∫c = − c + C
Transcendental Functions:
dx e∫ex
= x
+ C dx∫ax
= ax
ln(a)
+ C
dx n(x)∫ x
1
= l + C dx log (x)∫ 1
ln(a) x
1
= a + C
Inverse Trig Functions:
dx sin (x)∫ 1
√1−x2
= −1
+ C dx cos(x)∫ −1
√1−x2
= + C
dx tan (x)∫ 1
1+x2 = −1
+ C dx cot (x)∫ −1
1+x2 = −1
+ C
Integration By Parts (Product Rule):
dv uv du∫u = −∫v + C
Integration By Partial Fractions Example (Quotient Rule):
∫ dx
x(x+1) = ∫ x
Adx
+∫ x+1
Bdx
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