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99 POINT QUESTION, PLUS BRAINLIEST!!!
(Please answer genuinely, and do not answer just for points, if you do, your answer will be deleted, and those points you earned will be taken away...)
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THIS IS CALCULUS NOT BASIC MATH...

17.) The integral represents the volume of a solid. Describe the solid.
[you only have to answer A & B]
pi * (integral from 1 to 3) x^4 dx

a. What function is being revolved over what interval?
b. What is the axis of revolution?

** Please explain as much as possible to earn brainliest, 5 stars, thanks, and a certified answer...

Respuesta :

WojtekR
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.

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