Further integration (help)

[oscar24122000]

Hello, struggling with a question from further integration. The question is linked. Are we supposed to know how to graph this type of equation in an exam (guessing yes) and how would we graph it? also how would we do the question if y is squared same with x???
Help is appreciated, thanks!

Edit: The answer is 8/15 units squared but I do not know how to get to it?

[RuiAce]

Which textbook is that from? I'd definitely expect a 4U student to know how to graph it but not a 3U student.

(The relevant area integral is \(A = 2\int_0^1 x\sqrt{1-x}\,dx \), with the 2 arising from the symmetry of the resulting curve)

[oscar24122000]

Im not sure what textbook its from, I found it from a worksheet our 3u teacher gave us.

[RuiAce]

There's the old fashioned way of dealing with stationary points and what not to sketch \( y = x\sqrt{1-x} \) and \(y=-x\sqrt{1-x} \), but that's the only way I know of within the limitations of 3U to sketch such a curve. I'm highly positive that there are no "clever" or convenient ways of graphing this unless you insist on 4U techniques. If possible, please consult with your teacher regarding the source of the problem.

In the meantime, there's nothing wrong with using stationary points anyway, but it is very tedious. You may consider just using graphing software as an alternative.