91Ó°ÊÓ

Constant Force In Exercises, determine the work done by the constant force. The locomotive of a freight train pulls its cars with a constant force of 9 tons a distance of one-half mile.

Consider a beam of length \(L\). with a fulcrum \(x\) feet from one end (see figure). There are objects with weights \(W_{1}\) and \(W_{2}\) placed on opposite ends of the beam. Find \(x\) such that the system is in equilibrium. Two children weighing 50 pounds and 75 pounds are going to play on a seesaw that is 10 feet long.

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-1}^{1}\left[\left(1-x^{2}\right)-\left(x^{2}-1\right)\right] d x $$

Find the center of mass of the given system of point masses. $$ \begin{array}{|l|c|c|c|c|} \hline m_{i} & 12 & 6 & \frac{15}{2} & 15 \\ \hline\left(x_{1}, y_{1}\right) & (2,3) & (-1,5) & (6,8) & (2,-2) \\ \hline \end{array} $$

Hooke's Law, use Hooke's Law to determine the variable force in the spring problem. A force of 20 pounds stretches a spring 9 inches in an exercise machine. Find the work done in stretching the spring 1 foot from its natural position.

Find the area of the region by integrating (a) with respect to \(x\) and (b) with respect to \(y\). $$ \begin{aligned} &x=4-y^{2} \\ &x=y-2 \end{aligned} $$

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is 3 feet and the gasoline weighs 42 pounds per cubic foot.

Pumping Water A cylindrical water tank 4 meters high with a radius of 2 meters is buried so that the top of the tank is 1 meter below ground level (see figure). How much work is done in pumping a full tank of water up to ground level? (The water weighs 9800 newtons per cubic meter.)

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. \(y=x^{3}, \quad y=0, \quad x=2\) (a) the \(x\) -axis (b) the \(y\) -axis (c) the line \(x=4\)

Set up and evaluate the integrals for finding the area and moments about the \(x\) - and \(y\) -axes for the region bounded by the graphs of the equations. (Assume \(\rho=1 .\) ) $$ y=x^{2}-4, y=0 $$