Let R be the reglon enclosed by the line y=−1, the line x=2 and the curve y=x2−1.A solid is generated by rotating R about the line y=−1.Which one of the definite integrals gives the volume of the solid?Choose 1 answer:(A) π∫02x4dx(B) π∫−13(x2−1)2dx(c) π∫−13x4dx(D) π∫02(x2−1)2dx
Q. Let R be the reglon enclosed by the line y=−1, the line x=2 and the curve y=x2−1.A solid is generated by rotating R about the line y=−1.Which one of the definite integrals gives the volume of the solid?Choose 1 answer:(A) π∫02x4dx(B) π∫−13(x2−1)2dx(c) π∫−13x4dx(D) π∫02(x2−1)2dx
Identify Bounds of Integration: Identify the bounds of integration for the region R. The region is bounded by y=−1, x=2, and y=x2−1. The bounds for x are from 0 to 2.
Formula for Volume: Determine the formula for the volume of a solid of revolution using the washer method.The volume V is given by V=π∫ab(outer radius)2−(inner radius)2dx.
Calculate Outer Radius: Calculate the outer radius. The outer radius is the distance from the line y=−1 to the curve y=x2−1, which is (x2−1)−(−1)=x2.
Calculate Inner Radius: Calculate the inner radius. The inner radius is the distance from the line y=−1 to the line y=−1, which is 0.
Set up Integral for Volume: Set up the integral for the volume of the solid. V=π∫02(x2)2dx=π∫02x4dx.
Identify Correct Answer: Identify the correct answer from the given options.The correct integral is π∫02x4dx, which matches option (A).
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