Problem
1
Points
19
2
19
3
11
4
8
SECTION:
5
16
INSTRUCTOR:
6
19
7
8
Total
100
M161, Test 2, Fall 2003
NAME:
You may use alulators. No formulas an be
stored in your alulators.
Sore
The density of water is 1000 kg=m3 and the weight density of water is 62:5 lb=f t3. The aeleration due to
gravity is 9:8 m=se2 or 32 f t=se2 .
Error Bounds. Suppose jf (x)j
Midpoint Rules, then
00
K for a x b.
b a)
jET j K (12
n
2
3
If ET and EM are the errors in the Trapezoidal and
and jEM j K (b
a)3
24n2
:
Suppose jf (4) (x)j K for a x b. If ES is the error involved in using Simpson's Rules, then
(b a)
jES j K180
n
4
1
5
:
1. (a) Sketh the urve of the polar equation r = 4
sin().
0
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5
−4
−3
−2
−1
0
x
1
2
3
4
(b) Find an equation in Cartesian oordinates for the urve given in part (a).
() Find the area enlosed by the polar urve given in part (a).
2
5
2. (a) Sketh the urve of the parametri equations x = 4 os3 (), y = 4 sin3 (), 0 =2.
0
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5
−4
−3
−2
−1
0
x
1
2
3
4
5
(b) Calulate the length of the urve given in part (a).
() Find the area of the surfae obtain by rotating the urve given in part (a) about the y -axis.
3
3. (a) Sketh the urve of the parametri equations x = t3 3t2 , y = t3
diretion in whih the urve is traed as the parameter t inreases.
3t and indiate with an arrow the
0
5
4
3
2
1
0
−1
−2
−3
−4
−5
−5
−4
−3
−2
−1
0
x
1
2
3
4
5
(b) Find the points on the urve given in part (a) where the tangent line is horizontal or vertial|and draw
these tangents on your plot of the urve.
4
4. Find the area under one arh of the yloid x = t
sin(t), y
5
=1
os(t),
0 t 2 .
5. How
Z 2 large should we take n in order to guarantee that the Trapezoidal and Simpson's rule approximations
1
for
dx are aurate to within 0:0001?
1
x
6
6. A large tank is designed with ends in the shape of a trunated parabola: the region between the urves
y = x2 =2 and y = 12, measured in meters. The problem is to nd the hydrostati fore on one end of the
tank if it is lled to a depth of 8 meters with gasoline (assume that the density of gasoline is 800 kg=m3 ).
(a) On the piture inluded below, insert an axes and draw an appropriate arbitrary approximating retange
R (dierential element).
16
14
Top of tank
12
10
Gasoline level
8
6
4
2
0
−4
−3
−2
−1
0
x
1
2
3
4
(b) Find the area of the dierential element R and the fore on the dierential element R.
() Approximate the hydrostati fore against the end of the tank by a Riemann sum. Then let n approah
innity to express the fore as an integral and evaluate it.
7
7. Derive the trapezoidal rule formula for approximating the integral
plot of f given below that illustrates your derivation.
y
y=f(x)
a
b
x
8
b
Z
a
f (x)dx.
Inlude a drawing on the