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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: mano91 on July 11, 2009, 10:35:23 pm

Title: volumes of solids of rev.
Post by: mano91 on July 11, 2009, 10:35:23 pm: exercise 8D from essentials book.
question 29.

y=loge(x) between x=2 x=3 rotated about the x axis.
find the volume correct to 3 dec. places.

can this be done without a calculator?
ive tried looking at the inverse function and breaking it up into cylinders. but there is this tiny area i dont know how to do.
Title: Re: volumes of solids of rev.
Post by: TrueTears on July 11, 2009, 10:41:21 pm: Here's a good trick [Kind of stepping into the territory of integration by parts, check it out in wiki, good to know]

$y = xlog_ex$

$\frac{dy}{dx} = log_e(x) + 1$

$\int log_e(x) + 1 dx = xlog_ex$

$\int log_e(x) dx = xlog_ex - \int 1 dx$

For this question, you don't need the +c
Title: Re: volumes of solids of rev.
Post by: QuantumJG on July 17, 2009, 02:22:10 pm: Quote from: mano91 on July 11, 2009, 10:35:23 pm
exercise 8D from essentials book.
question 29.

y=loge(x) between x=2 x=3 rotated about the x axis.
find the volume correct to 3 dec. places.

can this be done without a calculator?
ive tried looking at the inverse function and breaking it up into cylinders. but there is this tiny area i dont know how to do.

V = pi*int[(ln(x))^2]dx

this can be found without a calculator! But you need a little tertiary calculus.

V = pi*[x*(lnx)^2 - 2int(lnx)dx]

= pi*[x*(lnx)^2 - 2x(lnx - 1)]

V (2 -> 3) = pi*[3(ln3)^2 - 6ln3 + 6 + 4ln2 - 4 - 2(ln2)^2]

= pi*[3ln3(ln3 - 2) - 2ln2(ln2 - 2) + 2]

= 2.642 units cubed (correct to 3 decimal places)
Title: Re: volumes of solids of rev.
Post by: dino on July 19, 2009, 09:04:15 pm: isn't it easier to get the inverse, hence an exponential?

area = pi int(x)^2 dy?

or maybe I'm simplifiying things?
Title: Re: volumes of solids of rev.
Post by: dcc on July 19, 2009, 10:37:14 pm: The simplest you'll get it to in VCE is $\pi \int_{\log\left(2\right)}^{\log\left(3\right)} u^2 e^{u}\: du$ , which is still impossible. Use a calculator :).

It's not only easier, but you are less likely to make a mistake / confuse examiner / make a mistake.
Title: Re: volumes of solids of rev.
Post by: TrueTears on July 19, 2009, 10:38:59 pm: Quote from: dino on July 19, 2009, 09:04:15 pm
isn't it easier to get the inverse, hence an exponential?

area = pi int(x)^2 dy?

or maybe I'm simplifiying things?
You can but why would you when you can just integrate it.