Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.


x=2−secy,x=2,y=π/3, and y=0; about x=2

Assume f and g are continuous, with f(x) ≥ g(x) ≥ 0 on [a, b]. The region bounded by the graphs of f and g and the lines x=a and x=b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = −8x−3 on [−2, 6] (Use calculus.)

Leaky Bucket A 1-kg bucket resting on the ground contains 3 kg of water. How much work is required to raise the bucket vertically a distance of 10 m if water leaks out of the bucket at a constant rate of 1/5 kg/m? Assume the weight of the rope used to raise the bucket is negligible. (Hint: Use the definition of work, W = ∫a^bF(y) dy, where F is the variable force required to lift an object along a vertical line from y=a to y=b.)

For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.

R is bounded by y=1−x^3, the x-axis, and the y-axis.

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

x =2

Let f(x) = {x   if 0≤x≤2

      2x−2  if 2<x≤5

      −2x+18 if 5<x≤6. 

Find the volume of the solid formed when the region bounded by the graph of f, the x-axis, and the line x=6 is revolved about the x-axis.