Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.

y=x^2,  x=y^2 about y = - 6.

Answer 1

13.5

Explanation:

We are given that two curves and a line

`y=x^2,x=y^2` about y=-6

`x=√(y)`

Height=
`√(y)-y^2`

Radius=y-(-6)=y+6

Now, we find intersecting points of two curves

`y=(y^2)^2`

`y=y^4`

`y^4-y=0`

`y(y^3-1)=0`

`y=0`

`y^3-1=0`

`y^3=1\implies y=1`

Now, volume generated by rotating the region by using cylindrical shell method is given by

`V=\int_(a)^(b)2\pi(height)(radius)dy`

Using the formula

`V=\int_(0)^(1)2\pi(√(y)-y^2)(y+6)dy`

`V=\int_(0)^(1)2\pi(y^{(3)/(2)}+6√(y)-y^3-6y^2)dy`

`V=2\pi[(2)/(5)y^{(5)/(2)}+4y^{(3)/(2)}-(1)/(4)y^4-2y^3]^(1)_(0)`

`V=2\pi((2)/(5)+4-(1)/(4)-2)`

`V=13.5`