Explanation Theorem used: “Pappus’s theorem for volumes: If the plane region is revolved once about a line in the plane that does not cut through the region’s interior, then the volume of the solid it generates is equal to the regions area times the distance traveled by the regions centroid during the revolution. If ρ is the distance from the axis of revolution to the centroid, then V = 2 π ρ A . Pappus’s theorem for surface area: If an arc of a smooth plane curve is revolved once about a line in the plane that does not cut through the arc’s interior, then the area of surface generates the arc equals the length L of the arc times the distance traveled by the arc’s centroid during the revolution. If ρ is the distance from the axis of revolution to the centroid, then S = 2 π ρ L ”. Calculation: The square region with vertices ( 0 , 2 ) , ( 2 , 0 ) , ( 4 , 2 ) , and ( 2 , 4 ) is shown in the fig 1. According to the pappus’s theorem, surface area is defined as: S = 2 π ρ L Where, ρ is the distance from the axis of revolution to the centroid and L is the distance travelled by the arc’s centroid during the revolution

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Chapter 6.6, Problem 43E

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The lateral surface area and the volume of a right-circular cone by using the pappus’s theorem.

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Chapter 6.6, Problem 42E

Chapter 6.6, Problem 44E

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