# Thread: Help with "Rotating body" and with optimizing problem.

1. ## Help with "Rotating body" and with optimizing problem.

Hi, first I have to thank you guys for the help, I relly appreciate it.
I got two questions, and I have tried and got some results but I'm not sure if it's quite right or the methods im using are right.

Question 1

Solve the volume of the rotating body that arise when the area between the curve y = 1/sqrt (x (1 + x ^ 2)) x> 1 and x axis, rotates about the x axis one revolution.

Y = 1/sqrt(x(1+x^2))

π y^2

π 1/(x(1+x^2)) dx

Here I do intergration by the "chainrule"

π x^-1 * (1+x^2)^-1 dx =( [x^-1 * arctanx] + x^-2 * arctanx)π

I have trouble to find a primitive function to the last intergral so I do intergration one more time.

x^-2 * arctanx = [ x^1/-1 * arctanx] - x^-1 * (1+x^2)^-1

I add the two intergrations to each other

π x^-1 * (1+x^2)^-1 dx = [x^-1 * arctanx]+[x^-1 / -1 * arctan x] - x^-1*(1+x^2)^-1

x^-1 * (1+x^2)^-1 dx = [x^-1 * arctanx - x^-1 *arctanx] from 1 to positive eternity

π x^-1 * (1+x^2)^-1 dx = ([x^-1 * arctanx - x^-1 *arctanx])/2

π([x^-1 * arctanx - x^-1 *arctanx])/2 from 1 to positive eternity

Then I put in the two values, eternity and 1 and I get as result
π arctan 1

Question 2 Optimizing.

Diabetes can sometimes be treated by operating in a capsule that delivers insulin into the blood. A researcher examines a capsule type that produce insulin at a speed of 0:52 * t * e ^ (-0.89t) cm ^ 3/day where t is time in days. When the capsule emits at most per day and how much is it?

0.52te^(-0.89t) dt

e^(-0.89t) = u
-0.89t = lnu
t = lnu/(-0.89)
dt = 1/(u*-0,89) du

I put in the substitions

0.52 lnu/( -0.89) * u * (1/u * -0.89) du =

= 0.52 lnu /(0.7921) du =

(0.52/0.7921) 1 * ln u du = [ u * lnu] - u * (1/u)

(0.52/0.7921) 1 * ln u du = [ u * lnu] - [ u]

I reset the substition and set in the orginal values

(0.52/0.7921)( e^-0.89t * - 0.89t - e^-0.89t)

How and what should I do know to answer the questions?
When the capsule emits at most per day and how much is it?

Thanks for helping me

2. Do somebody know a method to control if it's right? It's hard by only a calculator.. Please people see if I'm done right, I have to hand it in tomorrow

3. Originally Posted by Swedishmath
Hi, first I have to thank you guys for the help, I relly appreciate it.
I got two questions, and I have tried and got some results but I'm not sure if it's quite right or the methods im using are right.

Question 1

Solve the volume of the rotating body that arise when the area between the curve y = 1/sqrt (x (1 + x ^ 2)) x> 1 and x axis, rotates about the x axis one revolution.

Y = 1/sqrt(x(1+x^2))

π y^2

π 1/(x(1+x^2)) dx
In Latex, that is $\pi\int_0^1\frac{dx}{x(1+ x^2)}$

Here I do intergration by the "chainrule"
I have no idea what you mean by that. The chain rule is a method for differentiating, not integrating.

π x^-1 * (1+x^2)^-1 dx =( [x^-1 * arctanx] + x^-2 * arctanx)π
You seem to be trying to use the "product rule" in reverse but that is not true. In general $\int f(x)g(x)dx$ is NOT [tex]f(x)\int g(x)+ g(x)\int f(x)dx[tex].

I have trouble to find a primitive function to the last intergral so I do intergration one more time.
Instead, use "partial fractions"- find numbers A, B, and C such that $\frac{1}{x(x^2+ 1)}= \frac{A}{x}+ \frac{Bx+ C}{x^2+ 1}$

x^-2 * arctanx = [ x^1/-1 * arctanx] - x^-1 * (1+x^2)^-1

I add the two intergrations to each other

π x^-1 * (1+x^2)^-1 dx = [x^-1 * arctanx]+[x^-1 / -1 * arctan x] - x^-1*(1+x^2)^-1

x^-1 * (1+x^2)^-1 dx = [x^-1 * arctanx - x^-1 *arctanx] from 1 to positive eternity

π x^-1 * (1+x^2)^-1 dx = ([x^-1 * arctanx - x^-1 *arctanx])/2

π([x^-1 * arctanx - x^-1 *arctanx])/2 from 1 to positive eternity

Then I put in the two values, eternity and 1 and I get as result
π arctan 1

Question 2 Optimizing.

Diabetes can sometimes be treated by operating in a capsule that delivers insulin into the blood. A researcher examines a capsule type that produce insulin at a speed of 0:52 * t * e ^ (-0.89t) cm ^ 3/day where t is time in days. When the capsule emits at most per day and how much is it?

0.52te^(-0.89t) dt
Why are you integrating? You are given a function that gives the rate at which the capsule emits and are asked to find the time when it is emitting the most. That is, it is asking for the maximum value of the function. Do you know how to find maxima and minima for functions? You start by differentiating, not integrating!

e^(-0.89t) = u
-0.89t = lnu
t = lnu/(-0.89)
dt = 1/(u*-0,89) du
I put in the substitions

0.52 lnu/( -0.89) * u * (1/u * -0.89) du = 0.52 lnu /(0.7921) du =

(0.52/0.7921) 1 * ln u du = [ u * lnu] - u * (1/u)

(0.52/0.7921) 1 * ln u du = [ u * lnu] - [ u]

I reset the substition and set in the orginal values

(0.52/0.7921)( e^-0.89t * - 0.89t - e^-0.89t)=
As I said before, this problem does NOT require integration, it requires differentiation. However, if you did want to integrate this function, it is probably simpler to use "integration by parts" directly, taking $u= t$, $dv= e^{-0.89t}dt$.

How and what should I do know to answer the questions?
When the capsule emits at most per day and how much is it?
Thanks for helping me

4. I tried to do

And I got that A = 1 B = -1 and C=0

that gave me

1/x - x/(x^2+1) and its hard to find a primitive function to the last part.. but i tried and did this

lnx - int x/(x^2+1)

lnx - int 2x/2(x^2+1)

lnx- 1/2* int 2x/(x^2+1)

lnx-ln(x^2+1)/2

It's wrong again :/

And on the other question. I aldready got the derivitave why should i dervitave again?

I put the given 0.52te^(-0.89t) dt

0.52te^(-0.89t) = 0
to find maximum and I get it's when t= 0 and thats wrong, if you look it up in the calculator its when t=1

Then I do primitive because I'm interested to find the amount of sugar that de capsule realses and that time.

5. Originally Posted by Swedishmath
I tried to do

And I got that A = 1 B = -1 and C=0

that gave me

1/x - x/(x^2+1) and its hard to find a primitive function to the last part.. but i tried and did this

lnx - int x/(x^2+1)

lnx - int 2x/2(x^2+1)

lnx- 1/2* int 2x/(x^2+1)

lnx-ln(x^2+1)/2

It's wrong again :/
Why wrong? What does that give you for the volume?

And on the other question. I aldready got the derivitave why should i dervitave again?
The problem asks you to find a maximum value. A maximum value of what?

I put the given 0.52te^(-0.89t) dt

0.52te^(-0.89t) = 0
to find maximum and I get it's when t= 0 and thats wrong, if you look it up in the calculator its when t=1

Then I do primitive because I'm interested to find the amount of sugar that de capsule realses and that time.
Integrating (finding the primitive) will tell you how much sugar, in total, is released over a given time span. But you don't want to know the total amount released. The problem asks you to find the time at which the sugar is released at the fastest rate.