How do I approach this question when solving it? (Part B) (Calc 2)

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[tired mango]

I mean, 8 is just logic

taking cross sections perpendicular to the x-axis, you get circles

hence $\dd{V}=A(x)\dd{x}=\pi(r(x))^2\dd{x}$, and it's clear from a picture that $r(x)=f(x)-0=f(x)$

[shadow hazel]

Omegabet_

[tired mango]

so $V=\int\dd{V}=\int_2^\infty\pi(f(x))^2\dd{x}$

[shadow hazel]

Omegabet_

[tired mango]

b is just do the computation

they tell you how to integrate it

$u=x^2+5$ gives $\int\frac{x}{(x^2+5)^2}\dd{x}=\frac{1}{2}\int\frac{1}{u^2}\dd{u}$

[shadow hazel]

Omegabet_

[tired mango]

cause they did a u-sub

and it's an improper integral

if you mean why b appeared

(which you can just say "why did the infinity become b"

they did a sub

you change everything to u's

then you've done u-sub wrong

$\int_2^b$ are $x$ bounds

[shadow hazel]

Omegabet_

[tired mango]

$x=2\to u=2^2+5=9$

[shadow hazel]

Omegabet_

[tired mango]

$x=b\to u=b^2+5$

[shadow hazel]

Omegabet_

[tired mango]

any definite integral

it's a fact of u-sub

not of improper integrals