Question

The region R bounded by the curve y²=2x and the line y=x. Find the volume of the solid formed when R is rotated completely about
(a) the x-axis
(b) the y-axis

the answer for (a) 4π/3
the answer for (b) 16π/15

Answer 1

If you have the answer, why do you ask?

Answer 2

Maybe they would like to check their answer or understand how to get there @OOOPS ?

Answer 3

i have tried solving it and i keep getting the same value my method is wrong somewhere.

Answer 4

Can you walk me through your process so I can show you where your problem might be? You seem to be fairly consistent and thorough if you're getting the same answer, which is one of the best qualities to have already. =)

Answer 5

ok i drew the graph using the two equations y^2=2x (parabola on the y axis) and y=x (straight line).

then i equated the two equations to find the point of intersection

Answer 6

x^2-2x=0
x(x-2)=0
intercepts x=0 and x=2

Answer 7

So far so good. =)

Answer 8

i then used the formula for volume generated between x=a and x=b integrate πy^2 dx

Answer 9

then i substituted the function y^2=2x and integrated it and here is where i get the answer

Answer 10

4pix^2/3. we have to substitute x=2 and the answer is 16Pi/3

Answer 11

Not quite yet, this would be only what you get if you revolve y^2=2x around the x-axis from 0 to 2. This will look like this: However what you want:

Answer 12

ok

Answer 13

See how we drilled out that cone inside it? We need to subtract this part.