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Let D be the region in the first quadrant of the plane inside the limaçon r =7+5cos(theta) Find the volume?
Let D be the region in the first quadrant of the plane inside the limaçon r = 7+5cos(theta) Find the volume of the solid with base D that lies under the surface z = r^(1)sin(theta)
1 Answer
- kbLv 78 years agoFavorite Answer
V = ∫∫ z dA
...= ∫(θ = 0 to π/2) ∫(r = 0 to 7 + 5 cos θ) (r sin θ) * (r dr dθ), via polar coordinates
...= ∫(θ = 0 to π/2) (1/3)r^3 sin θ {for r = 0 to 7 + 5 cos θ} dθ
...= ∫(θ = 0 to π/2) (1/3)(7 + 5 cos θ)^3 sin θ dθ
...= ∫(u = 12 to 7) (1/3)u^3 * (-1/5) du, letting u = 7 + 5 cos θ
...= ∫(u = 7 to 12) (1/15) u^3 du
...= (1/60)u^4 {for u = 7 to 12}
...= 3667/12.
I hope this helps!
