Let’s see if we can continue our Obvious Integral Spree. We’re going to calculate the area of a circle by integration.
Here we have a circle. It has radius and can be described as . From that we infer that .
As all four sectors of the circle are equal, we just take the integral for the first quadrant, then multiply it by .
As area is positive we get
That won’t work, (if you must know) that leads us to
save for calculation errors; and we will end up with multiple divisions by zero and as an encore.
Let’s look at the circle again. There’s another way to describe and , because we know that and . We can rewrite as . As we remember that we end up with . Now we can take the square root and substitute back:
Now we need to know how to integrate with respect to :
There we are.
We also have to realize that evaluating the integral with x from 0 to will now lead us to another interval.
As evaluating from 0 to now means evaluating from to or to which is from to 0.
So now we can rewrite the integral as follows:
Factor out the constant,
Next problem is finding the primitive function of . But we remember (I hope so) that
and if we get
Where were we? O, yes, let’s substitute:
Out with the constant
and we finally get to
And with that we’re done.