Question

@jigglypuff314 @michele_laino @pooja195 @johnweldon1993 @irishboy123

Answer 1

http://math.stackexchange.com/questions/1240661/how-to-find-volume-of-the-given-solid-analytically

does this help?

Answer 2

Can you take a screenshot? the link is blocked for me

Answer 3

I'll see what I can do. but before that, do we know what the answer should be?

Answer 4

no

Answer 5

Ok well to find the volume, we'll need to integrate the area of the triangles. Let's see if we can establish A(y) for each triangle

Answer 6

i'm assuming these are the cross-sections of a cone, right?

Answer 7

a slanted cone

Answer 8

Ok so it's an iscocleses triangle meaning the base and the height are the same.

the base leg is equal to 2x, and the height is also going to be equal to 2x.

meaning the area of our triangles are
\[\ A(x)=\frac{1}{2}(2x)(2x)=\frac{1}{2}(2x)^2\]

Answer 9

ah, okay. And then I integrate that?

Answer 10

\[A(x)=2(x)^2\]

solve x^2+y^2=1 for x, replace x with the equation solved for y, and integrate taht from -1 to 1

Answer 11

solving x^2+y^2=1 for x gives \[x=\sqrt{1-y^2}\]

and when we substitute into \[A(x)=2(x)^2\]we get

\[A(y)=2(1-y^2)\]

Answer 12

why?

Answer 13

why do we want it in terms of y? well the initial problem gives us y from -1 to 1 so it makes sense to integrate it with respect to y from -1 to 1

Answer 14

oh, i see

Answer 15

do we need pi at all?

Answer 16

so we want to \[\int A(y) dy \] from -1 to 1

Answer 17

no pi

Answer 18

k, so