M161, Test 2, Spring 2003
The density of water is 1000kg=m3 or 62:5lb=f t3.
1. Find an equation in Cartesian oordinates for the urve desribed by the polar equation r = 3 sin .
2. Find the area enlosed by one loop of the polar urve r = os(2).
3. (a) Set up two dierent integrals|one integrated with respet to x and the other integrated with respet
to y | to nd the length of the urve y 2 = 4x, 0 y 2. You do not have to evaluate the integrals.
(b) For the same urve onsidered in part (a), set up two dierent integrals|one integrated with respet to
x and the other integrated with respet to y |to nd the area of the surfae obtained by rotating the urve
about the y -axis. Inlude a drawing that shows the dierential surfae area.
p
4. (a) For the urve given by parametri equations x = t; y = 1
a Cartesian equation of the urve.
t, t
0, eliminate the parameter to nd
(b) Sketh the urve and indiate with an arrow the diretion in whih the urve is traed as the parameter
inreases.
25
20
15
10
5
0
−5
−10
−15
−20
−25
−5
−4
−3
−2
−1
0
x
1
2
1
3
4
5
5. For the urve given by the parametri equations x = os t, y = et , 0 t =2:
(a) Sketh the urve and indiate with an arrow the diretion in whih the urve is traed as the parameter
t inreases.
5
4
3
2
1
0
−1
−1
−0.5
0
0.5
x
1
1.5
2
(b) Find the area bounded by the urve given above, y = 1 and x = 0.
6. Set up the integral and use your alulator to evaluate that integral for determining the perimeter of the
ellipse
x2
4
+
y2
9
= 1.
7. Find the tangent line to the urve x = t3 ; y = 2t at the point (x; y ) = (1; 2).
8. Find the hydrostati fore on one end of a ylindrial drum with radius 3ft. if the drum is submerged in
water 10 ft deep.
(a) On the piture inluded below, draw an appropriate arbitary approximating retangle R (dierential
element).
(b) Find the area the dierential element R and the fore on the dierential element R.
() Summing the fores over all of the approximating retangles and letting n ! 1, nd an integral expression
for the fore ating on the end of the ylindrial drum.
9. Derive the formula for determining the length of a urve for a urve given in parametri form by x =
f (t); y = g (t), a t b. Inlude a drawing on the plot of the urve given below that illustrates your
derivation. (For your drawing assume that t = a is assoiated with the origin and t = b is assoiated with
the point ( 2; 12).
2
x2+(y−3)2−9 = 0
12
water line
10
8
y
y
x2+(y−3)2=9
6
10 ft
4
2
x
0
−6
−4
−2
0
x
2
4
6
12
10
y
8
6
4
2
0
−6
−4
−2
0
x
2
3
4
6