Question

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Answer 1

You asked a similar question before, but the rotation was around the x-axis.
There's a little rule: \(\bbox[lightblue,2px]{the~axis~determines~the~function}\).

That means, the function \(f\) to use in the formula
\[\int_a^b \pi f^2(x)\,dx\] is not necessarily \(f(x)=x^2\).

Answer 2

You find \(f\) like this: \(\bbox[2px,lightblue]{ƒ~is~the~distance~between~the~curve~and~the~axis~of~rotation}\)

Answer 3

In your case, rotation is around y-axis, you must determine the function \(f(y)\):
Do this using the relation defining the curve: \(y=x^2\). In practice, it means: solve for \(x\) -> isolate \(x\) -> \(y=x^2\iff x=\dotsb\).

Answer 4

i dont really get it do i do: 1=x^2 -> x=1,-1?

Answer 5

The information `bounded by the graphs y = x^2, x = 4, and y = 1` means you must consider this region:

But.. now I must draw it, I don't understand the statement.. the three boundaries `y=x^2`, `x=4`(vertical line) and `y=1` (horizontal line) , the location of `x=4` seems awkward.. are you sure about this? is it not `x=1`?