Consider a right circular cone with radius r and height h. Find the volume of th

Question

Consider a right circular cone with radius r and height h. Find the volume of the cone (a) using slicing (cross-sections); and (b) using cylindrical shells.

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Explanation / Answer

sol: suppose that, 1. The length of a slice of the pyramid at height h (0 = h = 480) = 756(1 - h/480). The area of a slice of the pyramid at height h (0 = h = 480) = 756^2(1 - h/480)^2. Thus the volume of the pyramid = integral (h = 0 ---> 480) 756^2(1 - h/480)^2 dh so , = 756^2 integral (h = 0 ---> 480) (h/480 - 1)^2 dh = 756^2 * 480/3 * (h/480 - 1)^3 | (h = 0 ---> 480) = 756^2 * 480/3 * [0 - (- 1)] = 756^2 * 480/3 so , Substituting A = 756^2 and h = 480 into V = (1/3)Ah, we get V = (1/3) * 756^2 * 480 which is the same value as that obtained by the integral. let, 2. Consider a slice through the wedge in a direction parallel with the base diameter and the axis of the cylinder. Let the distance of the slice from the diameter and axis be h (0 = h = 2). Then its width = 2 sqrt (4 - h^2) and its depth = h (since the angle of the chop was 45°). So its area = 2h sqrt(4 - h^2) and the volume of the wedge = integral (h = 0 ---> 2) 2h sqrt(4 - h^2) dh = (- 2/3)(4 - h^2)^(3/2) | (h = 0 ---> 2) = (- 2/3)[0 - 4^(3/2)] = 16/3 = approx. 5.33 cubic inches so, I think this is correct but I'm not absolutely sure. The absence of a p term is unusual but not unprecedented. Compare this result with the volume of the corresponding cylinder (of depth 2") which is 8p or approx. 25.13 cubic inches. answer

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