Question

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
b) Sketch the region
c) Sketch the solid, and a typical disk or washer.

Answer 1

V = 34,13*π cubic units

Step-by-step explanation: See Annex

We find the common points of the two curves, solving the system of equations:

y² = 2*x x = 2*y ⇒ y = x/2

(x/2)² = 2*x

x²/4 = 2*x

x = 2*4 x = 8 and y = 8/2 y = 4

Then point P ( 8 ; 4 )

The other point Q is Q ( 0; 0)

From these two points, we get the integration limits for dy ( 0 , 4 )are the integration limits.

Now with the help of geogebra we have: In the annex segment ABCD is dy then

V = π *∫₀⁴ (R² - r² ) *dy = π *∫₀⁴ (2*y)² - (y²/2)² dy = π * ∫₀⁴ [(4y²) - y⁴/4 ] dy

V = π * [(4/3)y³ - (1/20)y⁵] |₀⁴

V = π * [ (4/3)*4³ - 0 - 1/20)*1024 + 0 )

V = π * [256/3 - 51,20]

V = 34,13*π cubic units