Question

A.y=flexside is revealed and the resort=0x** Mand the volume B Find the surface area ify = fx)=x OSXS 2. is revolved around the exis

Answer 1

3) We have given y=f(x)=xsin(x^3) and x=0 to (pi)^1/3

by using shell method

we know that the volume of the solid around y-axis is integration of (x=a to b)((2pi)*r*h)dx

we have radius r=x and height h=xsin(x^3) and thickness dx

So Volume V= integration of (x=0 to (pi)^1/3)((2pi)*x*xsin(x^3) )dx

=integration of (x=0 to (pi)^1/3)((2pi)x²sin(x^3) )dx

substitute u=x^3,du=3x^2dx implies x^2dx=du/3

=(2pi)* integration of (x=0 to (pi)^1/3)(sin(u))(du/3)

=(2pi)/3 *integration of (x=0 to (pi)^1/3)(sin(u))du

=(2pi)/3*[-cos(u)] from x=0 to (pi)^1/3

resubstitute u=x^3

=(2pi)/3*[-cos(x^3)] from x=0 to (pi)^1/3

=(2pi)/3*[-cos(((pi)^1/3)^3)-(-cos(0))]

=(2pi)/3*[-cos(pi)+1]

=(2pi)/3*[1+1]

=(4*pi)/3

Volume V=(4*pi)/3