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MATH 2374 Spring 2011 Worksheet 9

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MATH 2374 Spring 2011 Worksheet 9
MATH 2374
Spring 2011
Worksheet 9 12
1. Green’s Theorem can be used to calcualate the areas of some plane regions.
a) Prove
that the areaR of a plane
region D is equal to either one of the line integrals
R
R
1
x dy − y dx, − C y dx, C x dy, where C is the boundary of the region D oriented
2 C
counterclockwise.
b) Find the area enclosed by an ellipse with radii a and b.
Answer:πab.
c) The curve x = a cos3 t, y = a sin3 t is called astroid. Find the area enclosed by this
curve.
Answer: 83 a2 π.
d) Find the area enclosed by the curve y 2 = x2 − x4 .
Hint: To parametrize the curve, take y = xt, plug it into the curve equation
and then solve it for x and y in terms of t.
Mock midterm
1. Let r be the vector field defined by r(x, y, z) = xi + yj + zk. Compute
a) grad(|r| · div r); b) div(r × grad|r|); c) curl(|r|10 · r + 10 grad|r|).
Hint: The ’product rule’ formulas1 for grad, div and curl can be helpful. Also,
recall that curl(gradF) = 0 and div(curlF) = 0 for any vector field F.
r
Answer: a) 3 |r|
; b) 0; c) 0.
RR y
2. Evaluate the double integral
sin x dx dy, where D is the region bounded by the
D x
2
curves y = x, x = 1, y = 0.
Key words: iterated integral, bounds, order of integration.
1
Answer: 1−cos
.
2
3. The solid W is bounded by the surfaces x2 + y 2 = 4, 2x + z + 2 = 0 and lies above the
xy-plane. Find its volume.
Key words:√quadric surface, plane, intersection, triple integral, bounds.
.
Answer: 6 3 − 8π
3
2
2
4. A ring made of thin wire has the shape of
pthe circle x + y = 4x. The linear density of
the wire at point (x, y) is equal to ρ(x, y) = x2 + y 2 units. Find the total mass of the ring.
Key words: path integral, parametrization, double angle formulas.
Answer: 32 units.
5. Let C be the part of the parabola y = x2 between the points A(1, 1) and B(2, 4).
a) Set up (do not evaluate) the arc length integral for the curve C.
R
b) Evaluate the line integral C (x2 − 2xy) dx + (2xy + y 2 ) dy (assume that the path C
goes from A to B).
Key words: path integral, arc length, line integral, parametrization.
19
.
Answer: b) 40 30
6. A total of 2000 Joules of work was done by the force field F(x, y, z) = (xy 2 +x)i+(x2 y+3x)j
2
2
in moving a body of mass 1 kg around the ellipse x4 + y9 = 1, z = 0. How many full turns
around the ellipse did the body make?
Key words: line integral, closed curve, Green’s theorem, double integral.
Answer: 17.
1
See section 4.4 (p.306).
2
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