## Example 7.2.4(c): Applying the Substitution Rule

Of course everyone knows that the correct answer is*r*, but we want to

^{2}*compute*the answer, not just state it.

First we need to describe the circle mathematically. We know that
x is a circle,
which we can solve for ^{2} + y^{2} = r^{2}y:
where the positive root gives the upper half-circle (blue) and the negative one the lower part (red). |

Now we need to guess a change of variables that will make this integrand easier. Since we are dealing with a circle, the trig functions

*sin*and

*cos*might come to mind, and especially the fact that

*sin(t)*, or equivalently

^{2}+ cos(t)^{2}= 1Therefore we try the following substitution:r^{2}- (r cos(t))^{2}= (r sin(t))^{2}

With that substitution the integral is transformed into a much simpler version:x = r sin(t)so that

dx/dt = r cos(t)ordx = r cos(t) dt

where we determine the new integration interval

*[a, b]*later. Now we need to remember a few facts about trig functions, in particular the so-called "double-angle" formula:

Therefore our area iscos^{2}(t) = 1/2 (cos(2t) + 1)

where we have used the substitutionA = 2 r^{2}/2 cos(2t) + 1 dt = r^{2}[ cos(2t) dt + 1 dt ] =

= r^{2}[1/2 ( sin(2b) - sin(2a) ) + (b - a)]

*u = 2t*in our head to evaluate the first integral.

It remains to find the values of *a* and *b*. They originate
from the original substitution of *x = r sin(t)*. Therefore
*x = -r* must correspond to *a* such that *r sin(a) = -r*
and *x = r* must correspond to *b* where *r sin(b) = r*:

Therefore-r = r sin(a)andr = r sin(b)

*a = -/2*and

*b = /2*. Now we can compute the final answer:

Lucky us, we got the correct answer! Note that this time our change of variables is different from our previous examples:A = r^{2}[1/2 ( sin(2b) - sin(2a) ) + (b - a)] =

= r^{2}[1/2 ( sin() - sin(-) + ] =

= r^{2}

But as long as we correctly transform the

- usually we change an
expressioninxto asingle variableu.- this time we changed a
single variablexto anexpressionin another variablet.

*dx*term based on our substitution, anything goes.