1. Use the cylindrical shell method to find the volume of the solid obtained by rotating the region bounded byy=2x2and y=x3about the line x=−1.
2. A water tank has the shape of the lower half of a sphere of radius 10 feet. It is filled with water to a depth of 9 feet. Water is being pumped out from the tank through its top. Find how muchwork has been done when the depth of the water remaining in the tank is 2 feet. The weight density of water is 62.5 lb. per cubic foot.
3. Given secθdθ=ln∫secθ+tanθ+C, use integration by parts to computesec3θdθ∫. Make sure you show all intermediate steps of your calculation.Note: You will not earn any credit by directly using a reduction formula.
4. Use Simpson’s Rule with n=6subintervals to estimate the volume of the solid obtained by rotating the region shown below about thex–axis.
5. (a) Show that 11+t201/x∫dt=11+t2x∞∫dtby using the substitution u=1t.(b) Use the result in (a) to compute 11+t20x∫dt+11+t201/x∫dt
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