solve this Math questions

Question

The region R in the first quadrant is bounded by y=6sin(πx/2), y=6(x-2)^2 and y=4x+2, and contains the point (2,6)

The volume of the solid generated by rotating R about the x-axis is:

The volume of the solid generated by rotating R about the y-axis is:

The volume of the solid with R as its base and square cross-sections perpendicular to the x-axis (and to the xy-plane) is:

Answer 1

All 3 graphs passing throut point (1,6) and 2 graphs passing point (2,0). So we have for region R:

1 ≤ x ≤ 2, 6sin(πx/2) ≤ y ≤ 4x + 2

2 ≤ x ≤ 11/3, 6(x - 2)² ≤ y ≤ 4x + 2

Rotation about x-axi by disk metod: v = π∫₁²[(4x + 2)² - 36sin²(πx/2)]dx + π∫₂^11/3[(4x + 2)² - 36(x - 2)⁴]dx =

148.7 + 659.35 = 808.05 I use TI-84 to evaluate integrals

Rotation about y-axis, by Shell method: V = 2π∫₁²x[4x + 2 - 6sin(πx/2]dx + 2π∫₂^11/3x[4x + 2 - 6(x - 2)²]dx =

44.77 + 216.23 = 260.00

Now volume if R is base V= ∫₁²(4x + 2 - 6sin(πx/2))²dx + ∫₂^11/3(4x + 2 - 6(x - 2)²)²dx = 26.39 + 117.28 = 142.67;

All integrals evaluated by TI-84. I think that goal of this problem to setup these integrals