Lecture 5:
MATH 138W12003
January 13, 2012
I) Trigonometric substitution, Section 7.3
In this lecture we focus on computing integrals involving functions that include
√
a
2

x
2
,
√
x
2

a
2
and
√
x
2
+
a
2
using trigonometric substitutions. For example the first occurs when we want to
compute the area of an ellipse.
Example
If the equation of an ellipse is
x
2
/a
2
+
y
2
/b
2
= 1
then we can write the are as,
A
= 4
Z
a
0
b
a
√
a
2

x
2
dx.
At the moment we do not have the techniques for evaluating this. The new technique that we be
gin to develop is to make a trigonometric substitution of the form,
x
=
a
sin
θ
and
dx
=
a
cos
θdθ
.
The textbook calls this an
inverse substitution
because we are making this more complicated but
it is still a substitution so that’s the terminology I’m going to use.
There is a geometric interpretation of this substitution. We are defining a triangle that has one
side
x
, another
√
a
2

x
2
and a hypotenuse of
a
. We can assume that the angle we are dealing
with is in the range

π/
2
≤
θ
≤
π/
2
. If
x
is positive than so is the angle but if
x
is negative, then
the angle is negative. It is important to recall that
cos
θ
is not negative in this interval so that

cos
θ

= cos
θ
.
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 Fall '07
 Anoymous
 Calculus, Trigonometry, Integrals, tan θ, a2 − x2

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