Lecture5 - Lecture 5 MATH 138-W12-003 I Trigonometric...

Lecture 5: MATH 138-W12-003 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 .