Question

99 POINT QUESTION, PLUS BRAINLIEST!!!

Answer 1

1.Disc method.
In this method the volume is given by:

$\boxed{V=\pi\int\limits_a^b\big[f(x)\big]^2}$

so:

$V=\pi\int\limits_1^3x^4\,dx=\boxed{\pi\int\limits_1^3\big[x^2\big]^2\,dx}$

A) Function $f(x)=x^2$ over the interval $[1,3]$
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:

$\boxed{V=2\pi\int\limits_a^bx\cdot f(x)\,dx}$

So there will be:

$V=\pi\int\limits_1^3x^4\,dx=\dfrac{2}{2}\cdot\pi\int\limits_1^3x^4\,dx=2\pi\int\limits_1^3\dfrac{x^4}{2}\,dx= \boxed{2\pi\int\limits_1^3x\cdot\dfrac{x^3}{2}\,dx}$

A) Function $f(x)=\dfrac{x^3}{2}$ over the interval $[1,3]$
B) We use shell method and f(x) is function of variable x, so we rotate the curve about the y-axis.