Calculating Volume of Solid Rotated about Y-axis from Bounded Curves

[mxthuy95]

Homework Statement

Consider the region bounded by the curves y= lnx and y=( x-3)^2

Find the volume of the solid obtained by rotating the region about the y-axis

Homework Equations

The Attempt at a Solution

For this I solve for the x so i got x= e^y and x= (y)^(1/2) +3

and for volume equation i got :
1.47
∫ ((y)^(1/2) +3)^2 - (e^y)^2 = 12.481(pi). My answer is
0

different from the book. So can someone point out wat i done wrong and help me to get the right answer

Thank you

 

[Simon Bridge]

I cannot see what you did wrong because you did not show your working - you went right from the integral to the final numerical value ... and the integration you show does not look like the one for volume of rotation.

So I cannot tell what you were doing.
I suspect you need to sketch the two curves in the x-y plane and shade in the region bounded by them. The curve $y=(x-3)^2$ intersects the curve $x=e^y$ in two places. $x=\sqrt{y}+3$ only gives you part of the curve you need.

 

[haruspex]

You appear to be using the 'washer' method, i.e. a stack of elemental discs. This isn't going to work as a single range of integration because there's a sudden change of behaviour at the lower solution of ln(x) = (x-3)². (Try drawing the region, as Simon advises.) I suggest the cylinder method.

 

[mxthuy95]

Hi

Is this the right set up for the shell method

radius = x and height = lnx- (x-3)^2 so i get

A= 2(pi)x((x)(lnx-(x-3)^2) = 2(pi)(xlnx - (x)(x-3)^2 )so volume =

4.170
2(pi) ∫ (xlnx - (x)(x-3)^2
2.127

Thanks

 

[haruspex]

4.170
2(pi) ∫ (xlnx - (x)(x-3)^2
2.127

Yes, except that I get slightly different bounds: 2.130, 4.198.