Math 1210-001 Lab 9
Spring 2016; due April 7 at the beginning of lab
Name _____________________________________________UID______________________________
This lab contains section 4.1-4.4 material, mainly from 4.1-4.2. There will be a parallel WebWork
assignment to give you more opportunity for practice, especially for the section 4.3-4.4 material.
1) (15 points) Look for the pattern in order to write the following sum using summation notation
2
4
8
1024
1K
C
K
C .... C
.
3
5
7
21
(We don't care about the actual numerical value of the sum.)
1a) Start the index (counter) at i = 1. (Each term in the sum will be expressed as depending on i.)
1b) Start the index (counter) at k = 2. (Each term in the sum will be expressed as depending on k.) You
can get the answer from your answer to 1a by relating k to i.
10
2) (10 points) Suppose
10
> a = 15 and > b =K5.
i=1
i
i=1
i
Use the algebra for sums and multiples of sums to
calculate
10
>2 a C 3 b C 5
i=1
(Here we do care about the final numerical value).
i
i
3) (10 points) Consider the sum
10
>
2 kC5
kK3 .
k=1
Expand the general term in the sum, and then use the "magic" formulas below to compute the numerical
value for this sum. (Dapeng or Sarah might explain why the second one is true; the text explains where all
of the come from.) The final answer you should get is:
10
>
>
2$k C 5 $ k K 3 ;
k=1
565
(1)
>
n
> 1 = 1 C 1 C 1 C...C 1 = n
i= 1
n
> i = 1 C 2 C 3 C...C n = n n C2 1
i= 1
n
>i
2 = 1 C 22 C 32 C...C n2 =
i= 1
n nC1 2 nC1
6
n
>i
3 = 13 C 23 C 33 ...C n3 =
i= 1
n
>i
i= 1
4 = 14 C 24 C 34 C...C n4 =
n nC1
2
2
n n C 1 2 n C 1 3 n2 C 3 n K 1
30
4) (20 points) Find the total area of the inscribed and circumscribed rectangles between the graph of
1
f x = 2 x2 K
x C 1 and the x K axis on the interval 2, 4 with 4 equally spaced subintervals.
2
4a) Inscribed (left endpoints): (You don't need the picture to do the algebra, but it's nice to visualize.)
First write down the expression for the sum, then verify that its decimal value is 30.75
4b) Circumscribed (right endpoints). First write down the expression for the sum, then verify that its
decimal value is 42.25.
5) (30 points) In this problem you will evaluate the definite integral for the function and interval in
problem 4. In other words, for
1
f x = 2 x2 K
xC1
2
a, b = 2, 4
you will (choose to) partition the original interval into n equally spaced subintervals, use the right
endpoints to evaluate the function, and take the limit as n/Nof the Rieman sums:
b
n
f x dx = n lim
/N
>f
i=1
xi Dx.
a
The general template with n equal subdivisions is:
Since the total length of the interval a, b is b K a, each subinterval has width
bKa
Dx =
.
n
The sub-interval endpoints are
x0 = a
bKa
x1 = a C Dx = a C
n
2 bKa
x2 = a C 2 Dx = a C
n
bKa
xi = a C i Dx = a C i
.
n
The first interval is
I1 = x0 , x1
and the ith interval is
Ii = xi K 1 , xi .
5a) For
1
xC1
2
a, b = 2, 4
f x = 2 x2 K
and n equal length subintervals, Find D x.
5b) Find the formula for the right-hand endpoint xi of the ith subinterval Ii = xi K 1 , xi .
5c) Find the Riemann sum
n
>f
i=1
xi Dx .
Use the formulas included with problem 3 to simplify this Riemann sum so that it depends on n alone.
5d) Compute the definite integral (which should be the exact area under the graph):
4
n
1
2x K
x C 1 dx = n lim
f xi Dx.
/N
2
i=1
2
2
>
The amazing (second) Fundamental Theorem of Calculus says that definite integrals - computed by the
limit of Riemann sums method you just worked through in problem 5 - can actually be computed much
more simply using antidifferentiation. The Greeks could do the sorts of computations you worked through
in problem 5, but never came up with the ideas of derivatives and antiderivatives, which had to wait until
Newton and Leibniz.
Definite integrals have many more applications than just area (see Chapters 4-5), so this is an extremely
important fact. A precise statement of the FTC is that if f x is continuous on a, b then
b
n
f x dx d n lim
/N
>f
i=1
xi Dx
a
can actually be computed by
b
f x dx = F b K F a
a
where F is any antiderivative of f. (If you change F x by an additive constant, it will just be canceled out
when you evaluate F b K F a .) This is the main content of section 4.4.
6a) (10 points) Verify that your answer in 5d is correct by using antidifferentiation and the Fundamental
Theorem of Calculus, i.e. compute
4
2 x2 K
2
1
x C 1 dx
2
as F 4 K F 2 for some antiderivative of f x = 2 x2 K
1
x C 1.
2
6b) (5 points) In class on Wednesday, using Tuesday's notes, we used Riemann sum limits to show that
1
x2 dx =
0
1
3
Use the Fundamental Theorem of Calculus to get the same answer much more quickly.