Find the volume of the solid whose base is the region in the first quadrant bounded by y=x^5, y=1 and the y-axis and whose cross -sections perpendicular to the y-axis are equilateral triangles.

Question

I don't know what the bounds for the integral would be when I try to calculate the volume. I put everything in terms of x and so I got the volume to be
V=1/8(40/7x^(7/5)-5/4x^(8/5)),
which then needs to be evaluated by the bounds. I am not sure if my process is right and help would be appreciated.

Answer 1

each triangle has a base of length x, making its area √3/4 x^2
since x = y^(1/5) we have
v = ∫[0,1] √3/4 y^(2/5) dy = √3/4 * 5/7 y^(7/5) [0,1] = 5√3/28

Answer 2

okay thank you so much for the help!

Answer 3

i just tried to hand this in on my homework site and it said it was incorrect

Answer 4

also I got that each triangle would have a sidelength of x=5root(y) because I was told to set it in terms of y.

Answer 5

how about writing that as (5/7) sqrt (3/4)
or maybe sqrt (25/196

Answer 6

I mean sqrt (75/196)

Answer 7

if you meant trying the answer as sqrt(75/196 that is wrong

Answer 8

I did it again and agreed with oobleck

Answer 9

( 5√3 ) / 28

Answer 10

It is so easy to get mixed up with top and bottom online :(

Answer 11

oh no that is okay. let me try it

Answer 12

yes that is correct. could you explain your process on how you solved the question so I can recheck my own work and see where I went wrong

Answer 13

you should know by now that the area of an equilateral triangle of side s is
(s^2 √3)/4
Break it into two 30-60-90 right triangles to see how.