1.Disc method.
In this method the volume is given by:
[tex]\boxed{V=\pi\int\limits_a^b\big[f(x)\big]^2}[/tex]
so:
[tex]V=\pi\int\limits_1^3x^4\,dx=\boxed{\pi\int\limits_1^3\big[x^2\big]^2\,dx}[/tex]
A) Function [tex]f(x)=x^2[/tex] over the interval [tex][1,3][/tex]
B) We use disk method and f(x) is function of variable x, so we rotate the curve about the x-axis.
2. Shell method.
In this case volume is given by:
[tex]\boxed{V=2\pi\int\limits_a^bx\cdot f(x)\,dx}[/tex]
So there will be:
[tex]V=\pi\int\limits_1^3x^4\,dx=\dfrac{2}{2}\cdot\pi\int\limits_1^3x^4\,dx=2\pi\int\limits_1^3\dfrac{x^4}{2}\,dx=
\boxed{2\pi\int\limits_1^3x\cdot\dfrac{x^3}{2}\,dx}[/tex]
A) Function [tex]f(x)=\dfrac{x^3}{2}[/tex] over the interval [tex][1,3][/tex]
B) We use shell method and f(x) is function of variable x, so we rotate the curve about the y-axis.
