Lemniscate
The curve described in polar coordinates by r2 = cos(2) is called a lemniscate.
a) For what values of does there exist such a point (r, )?
b) For what values of is r at its minimum length?
c) For what values of is r at its maximum length?
d) Us
Numerical Integration
Compare the trapezoidal rule to the left Riemann sum. The area of each trapezoid is calculated using twice as much information (left and right endpoints) as
the area of each rectangle. This leads one to expect that applying the trape
Area of a Smile
The region between the curves y =
Find the area of that region.
1
2
x2
1
2
and y = x4
1 is smile shaped.
Solution
We begin by drawing a sketch of the region whose area we are trying to nd. The
lower lip of the smile is described by y = x4
Estimating ln(5)
a) Use the mean value theorem and the fundamental theorem of calculus to nd
Z 5
1
upper and lower bounds on
dx.
x
1
Z 5
1
b) Compute
dx.
x
1
c) Does your answer to (a) provide a good estimate of the value of ln(5)?
Solution
a) Use the mea
Using Simpsons Rule for the normal distribution
This problem uses Simpsons rule to approximate a denite integral important in probability.
In our probability unit, we learned that when given a probability density function f (x), we may
compute the probabi
Integration Intuition
When calculating areas, its a good idea to check your answer against a rough
visual estimate of the regions area. For each graph shown below, select the
value thats closest to the shaded area.
y
5
4
3
2
1
0
1
0
1
2
1
Approximate area
Riemann Sum Practice
Use a Riemann sum with n = 6 subdivisions to estimate the value of
Z
2
(3x + 2) dx.
0
Solution
This solution was calculated using the left Riemann sum, in which ci = xi 1 is
the left endpoint of each of the subintervals of [a, b]. To
Volume of Revolution Via Washers
Problem: By integrating with respect to the variable y, nd the volume of the
solidp revolution formed by rotating the region bounded by y = 0, x = 4 and
of
y = x about the line x = 6.
Solution: This problem was solved in r
Probability Function
A Poisson process is a situation in which a phenomenon occurs at a constant
average rate. Each occurrence is independent of all other occurrences; in a
Poisson process, an event does not become more likely to occur just because
its be
Average Bank Balance
An amount of money A0 compounded continuously at interest rate r increases
according to the law:
A(t) = A0 ert
(t=time in years.)
a) What is the average amount of money in the bank over the course of T years?
b) Check your work by plu
Integral of x4 cos x
This problem provides a lot of practice with integration by parts.
Compute the integral of x4 cos x.
Solution
A single application of integration by parts simplies, but does not solve, this
integral. We must repeat integration by part
Summation
Compute the following sums:
a)
5
X
k2
k=1
b)
3
X
(2k)2
k=1
c)
4
X
( 1)n n
n=1
d)
5
X
2k
k=0
Solution
The most di cult part of these problems is interpreting the summation notation.
That will grow easier with practice.
a)
5
X
k2
k=1
5
X
k2
=
12 +
lim
x! 0
sin x
1 cos x
In this problem attempt to evaluate:
sin x
x! 0 1
cos x
lim
using approximation.
a) Substitute linear approximations for sin x and cos x into this expression. Can
you tell what happens in the limit?
b) Substitute quadratic approxima
Weighted Average
The centroid or center of mass of a planar region is the point at which that
region balances perfectly, like a plate on the end of a stick. The coordinates of
the centroid are given by weighted averages.
R
x dA
The x coordinate of the cen
SOLUTIONS TO 18.01 EXERCISES
Unit 5. Integration techniques
5A. Inverse trigonometric functions; Hyperbolic functions
5A-1 a) tan
1
p
3=
3
b) sin
1
(
p
3
)=
2
3
p
p
p
c)p = 5 implies sin = 5/ 26, cos = 1/ 26, cot = 1/5, csc = 26/5,
tan
sec = 26 (from tria
Integral of sin(x) + cos(x)
Consider the following integral:
Z
sin(x) + cos(x) dx.
0
a) Use what you have learned about denite integrals to guess the value of this
integral.
b) Find antiderivatives of cos(x) and sin(x). Check your work.
c) Use the additi
Exploring a Parametric Curve
a) Describe the curve traced out by the parametrization:
x
y
=
=
t cos t
t sin t,
where 0 t 4.
b) Set up and simplify, but do not integrate, an expression for the arc length
Z 4
ds
dt of this curve.
dt
0
Solution
a) Describe t
1. Compute the area between the curves x = y 2
4y and x = 2y
Let f (y) = y 2
4y = y(y
4). f (y) = 0 when y = 0 or y = 4.
Let g(y) = 2y
y 2 = y(2
y2.
y). g(y) = 0 when y = 0 or y = 2.
x
2
3
y
?
The graphs of f and g intersect at (0, 0) and one other point.
Practice with Denite Integrals
Use antidierentiation to compute the following denite integrals. Check your
work using the geometric denition of the denite integral, graphing and estimation.
Z 2
a)
x2 dx
0
b)
c)
Z
e
1
Z
1
dx
x
0
sin x dx
/4
Solution
Z 2
a)
Revolution About the x-axis
Find the volume of the solid of revolution generated by rotating the regions
bounded by the curves given around the x-axis.
a) y = 3x x2 , y = 0
p
b) y = ax, y = 0, x = a
Solutions
a) y = 3x
x2 , y = 0
We start by sketching the
Integration by Change of Variables
Use a change of variables to compute the following integrals. Change both the
variable and the limits of substitution.
Z 4
p
a)
3x + 4 dx
0
b)
c)
Z
Z
3
x2
1
/2
x
dx
+1
sin5 x cos x dx
0
Solution
Z 4
p
a)
3x + 4 dx
0
The
Logs and Exponents
a) Prove that for x > 1:
a
Z
1
1/x
1
dt =
t
Z
1
(1/x)a
1
dt.
t
b) Assume x > 1. What is the geometric interpretation of the result of part a?
c) What does this tell you about the area between the x-axis and the graph of
1
x over the int
Path of a Falling Object
A teenager throws a ball o a rooftop. Assume that the x coordinate of the ball
is given by x(t) = t meters and its y coordinate satises the following properties:
y 00 (t)
=
0
y (0)
y(0)
=
=
9.8 meters/second
0
5 meters.
a) Find an