Question

\( 10. y = tanx,y = x,x = \pi /3; about the y - axis \)

Short answer

\(\int_0^1 \pi \left( {{{\left( {2 - {x^2}} \right)}^2} - {{(2 - \sqrt x )}^2}} \right)\)

Step-by-step solution

To Find the x-axis equation

When we rotate a thin vertical strip as shown in figure, about the \(y\)-axis.

We get a cylindrical shell with inner radius \(x\)and outer radius \(x + dx\)The height of the cylindrical shell is \(tan(x) - x\)

Volume of the cylindrical shell is

\(dV = \pi {( Outer Radius )^2}(Height) - \pi {( Inner Radius )^2}(Height)\)

\(dV = \pi {(x + dx)^2}(tan(x) - x) - \pi {(x)^2}(tan(x) - x)\)

\(dV = \pi \left( {{x^2} + 2xdx + {{(dx)}^2}} \right)(tan(x) - x) - \pi \left( {{x^2}} \right)(tan(x) - x)\)

Final Answer

We assume \({(dx)^2} \approx 0\)

\(dV = \pi \left( {{x^2} + 2xdx + 0 - {x^2}} \right)(tan(x) - x)\)

\(dV = 2\pi x(tan(x) - x)dx\)

\(V = 2\pi \int_0^{\pi /3} x tan(x) - {x^2}dx\)

\(V = 2\pi \int_0^{\pi /3} x tan(x) - {x^2}dx\)