Another Obvious Offshoot

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 :

thus

or

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:

or

Factor out the constant,

Next problem is finding the primitive function of . But we remember (I hope so) that

and if we get

or

or

Where were we? O, yes, let’s substitute:

Out with the constant

and we finally get to

And with that we’re done.