Integration problem. Please help (1 Viewer)

abdog · Mar 15, 2013

The shaded region in the diagram is bounded by the curve y=square root x+1, the y-axis and the line y=2. Find the volume of the solid formed when the shaded region is rotated about the x-axis.

pokka · Mar 15, 2013

If you sketched the graph then you would see that you take the volume of the cylinder from x=0 to x=1 and minus the volume of the "bowl" under the y=sqrt(x)+1 curve to find the shaded volume.

abdog · Mar 15, 2013

Can you show me how please?

pokka · Mar 15, 2013

sorry for the dodgy diagram but you should get the idea...

$First we want to find the volume of the cylinder which is basically the area of the rectangle between $x=0$ and $x=1$ and $y=0$ and $y=2$ rotated around the $x$-axis. We can just find this by using the formula for volume of a cylinder:$\\\\$Volume$_{cylinder}=\pi r^2h=\pi\times 2^2\times 1=4\pi\\\\$Next we want to find the volume of the red area rotated around the $x$-axis (it is the area under the curve $y=\sqrt{x}+1$ between $x=0$ and $x=1$):\\\\Volume$_{red}=\pi\int_0^1(\sqrt{x}+1)^2 dx\\\\$Once you find what that is, you can find the shaded blue volume by minusing the cylinder's volume by the red volume.$

abdog · Mar 15, 2013

Oh, when I say square root x, I mean square root (x+1). Sorry for the trouble. May I ask you to do it again? Sorry.

pokka · Mar 15, 2013

I suggest you sketch the graph for yourself so you can see. It should be similar anyway except you're using y=sqrt(x+1) and now you're looking at the volume of the cylinder and the volume of area under the curve which is between x=0 and x=3 (the point of intersection occurs here) instead of between x=0 and x=1. We choose between x=0 and x=3 because the shaded area is bounded by these x-values.

Integration problem. Please help (1 Viewer)

abdog

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