44 1. INTEGRATION, AREA, AND INITIAL VALUE PROBLEMS
e height of the ith rectangle is .x
i
/
2
D .iw/
2
D
2i
n
2
. Summing the areas we get that:
A Š
n
X
iD1
W H D
n
X
iD1
2
n
2i
n
2
Let’s simplify this with the algebraic rules for sums.
n
X
iD1
2
n
2i
n
2
D
n
X
iD1
2
n
4i
2
n
2
D
n
X
iD1
8i
2
n
3
D
8
n
3
n
X
iD1
i
2
D
8
n
3
n.n C1/.2n C 1/
6
Use
P
i
2
formula
D
16n
3
C 24n
2
C 8n
6n
3
D
8n
3
C 12n
2
C 4n
3n
3
Now we have a formula for the approximate area with n rectangles – the approximation gets
better as n grows. is means that
Z
2
0
x
2
dx D lim
n!1
8n
3
C 12n
2
C 4n
3n
3
D
8
3
units
2
which is the same result we get if we do the integral in the usual way.
is is a very cumbersome method of computing integrals – not used in practice – but it shows
that there is a theory for integrals, just as there is for derivatives. e fundamental theorem is a
godsend. Imagine if you did not know that integrals and anti-derivatives were the same thing.
Every integral would be a limit of sums of rectangles (or some other shape).
1.4. INDUCTION AND SUMS OF RECTANGLES 45
Many integrals cannot be done symbolically – with formulas and algebra. e discipline of
numerical analysis studies how to use things like rectangle-sum approximations to get useful
values for integrals that cannot be calculated with pure calculus.
PROBLEMS
Problem 1.66 Compute the following sums. You may use the formulas you are asked to prove
in Problem 1.67.
1.
40
X
iD1
i
2
2.
60
X
iD20
i
3.
30
X
iD1
.
2i C 5
/
4.
100
X
iD1
1
2
.i C 2/
5.
101
X
iD61
i
6.
23
X
iD5
2
i
7.
28
X
iD14
i
3
8.
37
X
iD18
.2i 1/
Problem 1.67 Use mathematical induction to demonstrate that the following formulas are
correct.
1.
n
X
iD1
.2i 1/ D n
2
2.
n
X
iD1
1 D n
3.
n
X
iD1
i
2
D
n.n C1/.2n C 1/
6
4.
n
X
iD1
i
3
D
n
2
.n C1/
2
4
5.
n
X
iD0
2
i
D 2
nC1
1
6.
n
X
iD0
3
i
D
1
2
3
n
C
1
1
7.
n
X
iD0
x
i
D
x
nC1
1
x 1
46 1. INTEGRATION, AREA, AND INITIAL VALUE PROBLEMS
Problem 1.68 Explain why the area under a line y D mx C b can always be found without
calculus.
Problem 1.69 A formula for approximating
Z
2
0
x
2
dx
with n rectangles was computed in this section. For n D 4; 6; 8; 12; 20; and 50 rectangles, com-
pute the error of the approximation.
Problem 1.70 Find the formula for the sum of rectangles for y D x
3
to approximate the in-
tegral of y D x
3
from x D 0 to x D c. Having found the formula, find the integral by taking a
limit.
Problem 1.71 e sum of rectangles used in this section was based on the right side of the
intervals. How would an approximation that used the left side of the interval be different? Could
it still be used with a limit, to compute integrals? Explain.
2.5-2.5
-1
7
Problem 1.72 e approximation sketched above uses trapezoids instead of rectangles to ap-
proximate the area under the curve. e area of each trapezoid is the average of the height of
the points on either side of the interval. Write out the approximation formula for n trapezoids
instead of n rectangles. Make it as simple as you can.
1.4. INDUCTION AND SUMS OF RECTANGLES 47
Problem 1.73 Is the trapezoid method more accurate than the rectangle method? Explain or
justify your answer.
Problem 1.74 Approximate
0
2
x
2
dx
using 20 intervals using rectangles and trapezoids. e area of a trapezoid with left side of height
h and right side of height k and width w is
A D
1
2
w.h Ck/
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