Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

y=x, y=0, x=2, and x=4; about x=1

I have nooo idea how to do this, the answers at the back of the textbook has 2 washers drawn, one at the top and the other at the bottom. I find this really complicated and I'm desperate for help :(

Answer 1

i'm a bit puzzled about how this looks - do you have a drawing of the curves (lines)?

Answer 2

yes jimmyrep, i will attach the file right now :)

Answer 3

right

Answer 4

This is a simpler diagram of just the lines, without the actual picture of the solid.

Answer 5

sorry i87 - i have to go out for a while be back in 1 hour if question is still open I'll look at it then

Answer 6

no problem, thanks

Answer 7

ok - have you done any calculus

Answer 8

yes i have, not this type though, mine was introductory calculus

Answer 9

integration?

Answer 10

yes, iknow integration

Answer 11

my main issue here was coming up with the diagram of the solid (diagram 1)

Answer 12

like i was able to draw the graph, i just didnt know how the textbook ended up with a such a complex diagram

Answer 13

right - the volume of rotation is the sum of all the tubes with very small thiskness formed on rotation. The thickness is dx and their length is the various values of y

The general formula for integration of this is \[2\pi \int\limits_{b}^{a} xy dx\]

Answer 14

In this case the curve ( line in this case) is y =x and the limits are 3 and 1
so we have

INt 2pi x^2 dx
= 2pi[ x^3/3} between 1 and 3
= 2pi [ 3^3 / 3 - 1^3 / 3]
= 2pi * 26/3
= 52pi/3 i cubic units

= 54.45 cu units

Answer 15

ok, that makes sense :)

Answer 16

i know you took a huge chunk of ur time to help me out and i really appreciate that

Answer 17

hold on i87 - theres a mistake the integral is INT 2pi x(x+1) between 1 and 3 not as i said

Answer 18

oh, alright, thanks for the correction.