Definite Integration question

[Arrows]

Hi,
I have been studying calculus on the topic 'volume of solids of revolution' which basically goes over finding volumes through cylinders with the formula V= the integral of πy^2dx. But that doesn't matter as such; it's more of a quier over a question. "Find the volume of the solid of revolution formed when the curve y=x^2+2 is rotated around the x-axis from x=1 to x=2."

Now my first thought was to do the "inverse chain rule" or chain rule with integrals, but supposedly that is wrong as it is not a linear function. Instead, we were to expand it, getting the answer of 293π/15 u^3 (units cubed, as it is volume being measured). But using the inverse chain got you 27/2π. These are completely different answers.

So overall question: can someone explain why the inverse chain rule only works with linear functions in regards to this question?

Many thanks,
Arrows

 

[Steven G]

Can you share your work so we can see what is going on?

In the formula you wrote for the volume you don't have to use y for y. In this case, you can replace y with x^2 + 2 (after all, that is what y equals).
Just integrate (x^2+2)dx from x = 1 to x=2 and multiply that result by π

 

[blamocur]

Just integrate (x^2+2)dx

[imath](x^2+2)^\mathbf{2}[/imath] ?

 

[Steven G]

[imath](x^2+2)^\mathbf{2}[/imath] ?

Thank you!