Math 1320-6 Lab 2
T.A.: Kyle R Steffen
21 January 2016
Instructions and due date:
• Due: 28 January 2016 at the start of class.
• For full credit: Show all of your work (including any sketches), and
simplify your final answers.
• Work together! However, your work should be your own (not copied
from a group member).
1. Calculate the length of the curve defined parametrically by y = e−t cos(t), x = e−t sin(t)
from t = 0 to t = 2π.
Math 1320-6
Lab 2 - Page 2 of 6
2. A steady wind blows a kite due west. The height above the ground from horizontal
position x ≥ 0 is given by
3
1
1
y = (x + 1) 2 − (x + 1) 2 .
3
(a) Find the position where the kite falls onto the ground.
(b) Find the distance traveled by the kite before it fell on the ground. (Hint: 1 + (
is a perfect square.)
dy 2
)
dx
Math 1320-6
Lab 2 - Page 3 of 6
3. Consider a thin wire of length 3 cm, positioned so that it coincides with 0 ≤ x ≤ 3 and
y = 0. The wire has a density in mg cm−1 given by:
πx π
λ(x) = 2 + sin
3
3
(a) Draw the graph of the density function. Find the mass of the wire in mg. (Hint:the
mass can be viewed as the area of the region bounded by the density function.)
(b) Use the symmetry of λ(x) to guess the value of x̄ (i.e. to guess the location of the
center of the mass).
(c) Write down an integral expressing x̄ and evaluate the integral to check if it matches
with your guess from part (b).
Math 1320-6
Lab 2 - Page 4 of 6
4. Recall that there are three methods to estimate a definite integral: Midpoint Rule,
Trapezoid Rule, and Simpson’s Rule. Assume that we want to evaluate the integral of a
function f (x) on the interval [a, b]. First, we will divide the interval into n subintervals
b−a
with length ∆x =
. Then, we will estimate the area under the curve on each
n
subinterval Ai as following:
xi − xi−1
2
Ai = f (x̄i )∆x
Z b
f (x)dx ≈ ∆x[f (x̄1 )+f (x̄2 )+· · · +f (x̄n )]
x̄i =
a
1
Ai = [f (xi−1 ) + f (xi )]∆x
Z b 2
∆x
(A1 + A2 + · · · + An )
f (x)dx ≈
2
a
∆x
=
[f (x0 )+2f (x1 )+· · ·+2f (xn−1 )+f (xn )]
2
1
Ai =
[f (xi−1 ) + 4f (xi ) + f (xi+1 )]∆x
3
Z b
∆x
f (x)dx ≈
(A1 + A3 + · · · + An−1 )
3
a
∆x
=
[f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 )
3
+2f (x4 ) + · · · + 4f (xn−1 ) + f (xn )]
Math 1320-6
Lab 2 - Page 5 of 6
(a) Let n = 4, and use each of the three different methods to estimate
Z 2
cos(1 + x)dx.
0
Note: Keep your final answer to 4 decimal places.
(b) Calculate the integral and compare the result with the results from part (a). Which
method from part (a) gives the closest estimation?
Math 1320-6
Lab 2 - Page 6 of 6
5. Suppose that you are gathering water from a well by pulling a rope attached to a bucket
of water. As the force of gravity FG acts on the bucket, this requires a work W given by
Z d
W =
FG (y)dy
0
where d = 8 m is the depth of the well and y = 0 corresponds to the bottom of the
well. If the mass of the rope is negligible, the force of gravity on the bucket is given by
FG (y) = m(y)g, where g = 9.81 m s−2 and m(y) is the mass of the bucket.
(a) Suppose that the mass of the bucket is constant, m(y) = 5 kg. Find W .
(b) Now suppose that the bucket has the same initial mass m(0) = 5 kg, but is continuously leaking so that its mass is reduced by a constant rate of 0.25 kg for every
meter it is raised. Find m(y) and W .