Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

a. What is the radius of a cylindrical shell at a point x in [0, 2]?

Calculating work for different springs Calculate the work required to stretch the following springs 0.5m from their equilibrium positions. Assume Hooke’s law is obeyed.

a. A spring that requires a force of 50 N to be stretched 0.2 m from its equilibrium position

In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.

a. What is the SAV ratio of a cube with side lengths a?

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

a. The displacement between t=0 and t=5

Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.

a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].

21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = ln x, for 1≤x≤4

Determine whether the following statements are true and give an explanation or counterexample.

a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.