Question

Sketch the region bounded by the curves x = y² and 2x = y + 1 then use the shell method to find the volume of the solid generated by revolving this region about the y-axis.

a) 49π/20
b) 9π/20
c) 29π/20
d) 89π/20
e) 69π/20

Answer 1

The volume of the solid formed by revolving the region bounded by x = y² and 2x = y + 1 about the y-axis, calculated using the shell method, is 49π/20.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by x = y² and 2x = y + 1 around the y-axis, we first sketch the region. The intersection points of the curves can be found by setting x = y² equal to 2x = y + 1, solving for y. We find that y equals 0 and 1, which are our bounds for integration.

To use the shell method, consider a typical shell at position y with height 2x - y² and radius y. The volume of the shell with thickness delta y is approximately 2πy(2x - y²) delta y. Integrating this from y = 0 to y = 1 gives us the total volume:

V = ∫₀¹ 2πy(2y + 1 - y²) dy

After computing the integral, we find the volume to be 49π/20, which is answer choice (a).