Question

The region R is bounded by the x-axis, y-axis, x = 3 and y = 1/sqrt(x+1).

Answer 1

A. Find the area of region R.
B.Find the volume of the solid formed when the region R is revolved about the x-axis
C. The solid formed in part B is divided into two solids of equal volume by a plane perpendicular to the
x-axis. Find the x-value where this plane intersects the x-axis.

Answer 2

first lets graph the region https://www.desmos.com/calculator/2bz6ctgloa

Answer 3

Yeah I was able to do that much.

Answer 4

so the area of the region R , we take the integral
$$ \Large \int_{0}^{3} \frac{1}{\sqrt{x+1}}~dx$$

Answer 5

I got 2

Answer 6

thats correct so far

Answer 7

So 2 is the answer to A then right?

Answer 8

right

Answer 9

Alright, then how do we do part B?

Answer 10

part B we revolve about x axis
$$ \Large \int_{0}^{3} \pi \left( \frac{1}{\sqrt{x+1}} \right)^2~dx$$

Answer 11

pi ln (4)

Answer 12

thats correct

Answer 13

Now how do we do part c?

Answer 14

we need to find the x point that divides the volume in half

Answer 15

you can't just cut it in half, because the region is not symmetric

Answer 16

So then where do we cut it?

Answer 17

solve for c
$$\LARGE {
\int_{0}^{\color{red}c} \pi \left( \frac{1}{\sqrt{x+1}} \right)^2~dx = \frac{1}{2}\cdot \pi \ln 4

}
$$

Answer 18

pi ln (c+1)

Answer 19

Wait sorry I'm solving for c not integrating.

Answer 20

I got ln|x+1| - ln|1| = ln|4| - ln|x+1|

Answer 21

x=1

Answer 22

right, so we need to solve
ln| c + 1 | - ln| 0 + 1| = 1/2 * Pi * ln(4)

Answer 23

is it not x/c =1?

Answer 24

x/c means x or c as I put x when I solved.

Answer 25

$$
\Large {
\int_{0}^{c} \pi \left( \frac{1}{\sqrt{x+1}} \right)^2~dx
\\~\\=\int_{0}^{c} \pi \left( \frac{1}{x+1} \right)~dx
\\~\\ =\pi \left[ \ln |x + 1| \right]_{0}^{c}
\\~\\ =\pi ( \ln|c+1| - \ln|0+1|)
\\~\\ =\pi ( \ln|c+1| - \ln|1|)
\\~\\ =\pi \ln|c+1|
}

$$

Answer 26

ok so far?

Answer 27

Yeah

Answer 28

so we need to solve
pi ln ( c + 1) = 1/2 * pi * ln(4)

ln(c+1) = 1/2 * ln (4)

ln(c+1) = ln(4 ^(1/2) )
ln(c+1) = ln(2)

so c = 1

Answer 29

Alright so I guess I was right, just needed to see the work.

Answer 30

Thanks for the help.

Answer 31

your welcome :)