Turning points question

[Anthonyk2013]

Wondering what's going on in this part of the solution.

 

[MGM]

They took the 1/2 outside of the integration, and integrated the expression that was left, i.e (y+3)

y+3, when integrated w.r.t dy, becomes (y^2/2)+3y

 

[Anthonyk2013]

They took the 1/2 outside of the integration, and integrated the expression that was left, i.e (y+3)

y+3, when integrated w.r.t dy, becomes (y^2/2)+3y

Thanks i I see it now. But what happened to the x^2 on the above line.

 

[ksdhart]

Nothing happened to it. The solution in the image you posted derives a value for x² in terms of y:

\(\displaystyle x^2=\frac{y-3}{2}\)

Then take that value and plug it into the integral:

\(\displaystyle \displaystyle \text{Volume}_{y-axis}=\int _3^{21}\:\pi \:x^2dy=\pi \int _3^{21}\:\frac{y-3}{2}dy\)