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Assignment 8
This assignment is due at the beginning of class on Monday, May 5, 2008.
1.
(a) For each deﬁnite integral below, use
n
= 10 rectangles to approximate the area under the
curve. Please give answers to at least four decimal places.
i.
±
2
1
1
x
dx
ii.
±
4
1
1
x
dx
iii.
±
8
1
1
x
dx
(b) What do you notice about your answers, above?
(What is the pattern?)
Use the observed
pattern to guess at the value of
±
32
1
1
x
dx.
(c) Use the Fundamental Theorem of Calculus to explain
the pattern you observed in part (b).
2. We know, from an earlier assignment, that the derivative of
y
= tan

1
(
x
) is
y
±
=
1
1+
x
2
and so,
according to the Fundamental Theorem of Calculus,
±
1
1 +
x
2
dx
= tan

1
(
x
) +
C.
In this problem we solve the indeﬁnite integral
±
1
1 +
x
2
dx
directly, using a certain substitution.
Here is how:
The expression 1 +
x
2
should remind us of the Pythagorean Theorem. Draw a right triangle with
legs of lengths 1 and
x
. (What is the length of the hypotenuse?) Let
θ
represent the angle opposite
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This note was uploaded on 04/02/2009 for the course MTH 142 taught by Professor Staff during the Spring '08 term at Sam Houston State University.
 Spring '08
 STAFF
 Calculus, Angles

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