Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y2 = 2x, x = 2y; about the y-axis

Answer 1

With this problem, we can find the volume using the Washer method.

( Kinda tough typing this all out so excuse the length ;-) )

Here's our general equation for volume: V = ∫ π [(R(y)_outer)² _– (R(y)_inner)²] dy

***It may be helpful to graph the two equations to see which one is your outside function (R(y)_outer) and your inside function (R(y)_inner)

It turns out

1. (R(y)_outer) → x = 2y [Equation 1]
2. (R(y)_inner) → y² = 2x [Equation 2]
3. OR y²/2 = x (for later use when ready to integrate)

Next, we need to determine our definite integral bounds (where the 2 equations intersect)

1. Set equations equal by substituting Equation 1 into Equation 2 for x: y²= 2(2y)
2. Solve for y to determine your lower and upper bounds:
3. y² – 4y = 0 → y(y - 4) = 0
4. y = 0 and y = 4 {We now have our lower & upper bound}

Now we have our equation: V = ∫ π [ (2y)² _– (y²/2 )²] dy {Integral bounded from 0 to 4}

1. Can clean up expression a little: V = ∫ π [ 4y² _— (y⁴/4 )] dy

Integrate:

1. V = π [ (4/3) y³ — (1/20) y⁵ ] {bounded from 0 to 4}

Plug in upper bound = 4 into y and subtract from lower bound to get your value

1. Lower bound = 0 so it'll make entire expression 0

After plugging in 4 into y and simplifying:

1. V = π [ (256/3) — (256/5) ]

FINAL ANSWER:

1. V = π [512/15] ≈ 107.233