Question

Another stupid question, i'm trying to switch the order of integration.

Answer 1

so I have \[\int^2_0\int^{4x-x^2}_{2x}dydx\] and I just need to reverse the order of integration. The one that is throwing me off is writing \(y=4x-x^2\) in terms of x. Like I see that x goes from that curve to \(x=\frac{y}{2}\) and y goes from 0 to 4, but I can't figure out how my textbook is getting \[2-\sqrt{4-y}\]

Answer 2

huh?

Answer 3

still need help @richyw ?

Answer 4

when you get back and want to give it a try, use completing the square. The problem with the quadratic term is that it's not clearly bijectiv, therefore there is more than one solution when you try to inverse it.

Answer 5

yes I still need help!

Answer 6

So you are confused on how they obtained this result: \[ \Large x=2- \sqrt{4-y} \] ?

Answer 7

yup

Answer 8

oh wait can I just factor out an x and ten complete the square?

Answer 9

\[\Large y= -x^2+4x \\\Large -y = x^2-4x\]
Complete the square for the \(x\) term on the RHS of the equation.

Answer 10

\[ \Large 4-y= x^2-4x+4=(x-2)^2\]

Answer 11

\[\Large x= 2 \underbrace{\pm}_{\text{ non bijective}} \sqrt{4-y} \]

Answer 12

yes thanks a lot I know which one I need though

Answer 13

welcome