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The line segment joining the origin on the point (3,2) is revolved about the x-axis to generate a cone of height 3 and base radius 2. Find the cone's surface area using the parametrization x=3t, y=2t, on the interval (0,1)
A frustum of cone is made by rotating about Ox the area bounded by the lines y=x, x=0, x=a and x=b, with b>a. What is its volume?
Verifying a Formula
(a) Given a circular sector with radius L and central angle (see figure), show that the area of the sector is given by .
(b) By joining the straight-line edges of the sector in part (a), a right circular cone is formed (see figure) and the lateral surface area of the cone is the same as the area of the sector. Show that the area is , where r is the radius of the base of the cone. (Hint: The arc length of the sector equals the circumference of the base of the cone.)
(c) Use the result of part (b) to verify that the formula for the lateral surface area of the frustum of a cone with slant height L and radii r1 and r2 (see figure) is .(Note: This formula was used to develop the integral for finding the surface area of a surface of revolution.)