
Area under a curve
Find the area under the graph to the x axis from 2 to 2 for
f(x) = (4  x^2)^(1/2)
Okay, so I understand how to do the problem overall but I don't know how to take the antiderivative of that particular function. Is there an easy way to do the chain rule in reverse or what...

Is that $\displaystyle f(x)=\sqrt{4x^{2}}$? If so, I wouldn't integrate. You can solve the problem without integrating. Think about what the shape of that curve is.

Yeah, that's the function. It's a semicircle?

Quote:
It's a semicircle?
Show me that it's a semicircle.

It just is. What do you mean?

I asked my question because you seemed unsure of whether it's a semicircle or not. If you are sure, then what do you get for the area under it?

Oh, sorry. I had the question mark there because I didn't know where you were going with it, not whether or not it's a semicircle. Um...I don't really know. It's doesn't quite touch the xaxis according to my graphing calculator, and as a result it is sort of messing with all of the estimations I've attempted, and I don't know how to integrate a function like that.

Are you required to solve this problem using integration? Because I find that simply writing the answer down is a lot easier. What's the radius of this semicircle? And what's the area of a semicircle in general?

Follow Ackbeet's advice. Integration is just the area under the curve so...

I think integration is preferred, but this method should suffice.



Or, more accurately, $\displaystyle 2\pi.$

Touche, but it was supposed to be rounded to two decimal places, so I had to.

Then 6.28 might not be the more accurate answer, but it is the better answer. In engineering, at least, you better give an answer that your boss is happy with. If he wants decimals, give him decimals. Otherwise, it won't be as useful to him.