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.

d. A piecewise function for s(t)

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.

Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.

d. Write an integral for the volume of the solid.

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.

d. A piecewise function for s(t)

Displacement and distance from velocity Consider the velocity function shown below of an object moving along a line. Assume time is measured in seconds and distance is measured in meters. The areas of four regions bounded by the velocity curve and the t-axis are also given.

d. What is the displacement of the object over the interval [0, 8]? 

Compressing and stretching a spring Suppose a force of 30 N is required to stretch and hold a spring 0.2 m from its equilibrium position.

d. How much additional work is required to stretch the spring 0.2m if it has already been stretched 0.2m from its equilibrium position?

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function

$v(t)= \(\begin{cases}\)3 t & \(\text\) { if } 0 \(\leq\) t<20 \\ 60 & \(\text\) { if } 20 \(\leq\) t<45 \\ 240-4 t & \(\text\) { if } t \(\geq\) 45\(\end{cases}\)$

where is measured in seconds and v has units of m/s. 

d. What is the position of the automobile when t=75?

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.

d. How long does it take the racer to travel 300 ft?