Compute the area of a circle with radius rBack
Example 7.2.4(c): Applying the Substitution Rule |
Compute the area of a circle with radius r |
r2,
but we want to compute the answer, not just state it.
First we need to describe the circle mathematically. We know that
x2 + y2 = r2 is a circle,
which we can solve for 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)2 + cos(t)2 = 1, or equivalently
r2 - (r cos(t))2 = (r sin(t))2Therefore we try the following substitution:
x = r sin(t) so thatWith that substitution the integral is transformed into a much simpler version:
dx/dt = r cos(t) or dx = 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:
cos2(t) = 1/2 (cos(2t) + 1)Therefore our area is
A = 2 r2/2where we have used the substitution u = 2t in our head to evaluate the first integral.cos(2t) + 1 dt = r2 [
= r2 [1/2 ( sin(2b) - sin(2a) ) + (b - a)]
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:
-r = r sin(a) and r = r sin(b)Therefore a = -
/2 and
b = /2. Now we can compute the
final answer:
A = r2 [1/2 ( sin(2b) - sin(2a) ) + (b - a)] =Lucky us, we got the correct answer! Note that this time our change of variables is different from our previous examples:
= r2 [1/2 ( sin(
=
But as long as we correctly transform the dx term based on our substitution, anything goes.
- usually we change an expression in x to a single variable u.
- this time we changed a single variable x to an expression in another variable t.
