Question

Find the volume of the graph of x=sqrt(1+x) from y=-1 to 3 rotated around the y-axis?

Answer 1

might have to draw a graph to see what going on

Answer 2

Yeah...I'm not sure how to do that on here...I didn't think the question made much sense when I typed it like that.

Answer 3

x = sqrt(1+x) ? or is that a y in there someplace?

Answer 4

yes, sorry..it's x=sqrt(1+y).

Answer 5

which would be easier for this; shells or discs?

Answer 6

discs

Answer 7

this is the same result as if we were doing:

y = sqrt(x+1) from x=[-1,3] around the x axis right?

Answer 8

that is it, but it is being rotated around the y axis

Answer 9

the equation is the same regardless of where you put the variable in this case; the same shape results; but ill stick with the original all the same

Answer 10

now, what we want to determine is the area of a typical circle and then add them all up from -1 to 3

Answer 11

the area of a circle is pi r^2 right?

Answer 12

yeah...I have the equation as being \[\pi \int\limits_{-1}^{3}(\sqrt{y+1}^{2}dy?\]

Answer 13

thats correct; now the ^2 and the sqrt "cancel" which leaves us with:

pi. {S} (y+1) dy ; [-1,3] right?

Answer 14

yeah..and I solved that for what I thought was pi(y^2/2 + y) and from [-1,3] I thought it was 12 pi, but that is incorrect.

Answer 15

lets see ....

pi (9/2+3) - pi (1/2 - 1)

pi (9/2 +3 -1/2 +1)

pi (8/2 + 4) ... you see where this is headed?

Answer 16

being able to integrate is no good if you forget how to do the simple math afterwards ;)

Answer 17

we can save ourselves some mathical skills by lifting the whole thing up by 1

pi. {S} y dy ; [0,4], the 0 is useless ...

pi y^2/2 = pi 16/2 = 8

Answer 18

8 pi ....

Answer 19

I see what I did wrong...I made the first term 9/2 + 9 because I squared the second y...silly mistake

Answer 20

I like the idea of lifting it all up by one...hadn't though of that one. Thanks for the help.

Answer 21

youre welcome :)