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?
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 rect
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
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
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 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 th
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 l
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
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 d
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 ban
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 s
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
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
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 b
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
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 o
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 leng
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 gra
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
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
S
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
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 ar
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