91Ó°ÊÓ

An 8-ft-long trough has ends that are equilateral triangles with sides that are \(2 \mathrm{ft}\) long. If the trough is full of water weighing \(62.4 \mathrm{lb} / \mathrm{ft}^{3}\), find the work required to empty it by pumping the water through a pipe that extends \(1 \mathrm{ft}\) above the top of the trough.

Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. \(y=(x-1)^{2}, \quad y=x+1 ;\) the \(x\) -axis

Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. \(y=x^{2}, \quad y=2 x-1, \quad y=4 ; \quad\) the \(y\) -axis

(a) plot the graph of the function \(f\). (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to four decimal places. $$ f(x)=\frac{x}{1+x^{4}} ; \quad[0,1] $$

In Exercises 31 and 32, use differentials to approximate the arc length of the graph of the equation from \(P\) to \(Q\). $$ y=x^{3}+1 ; \quad P(1,2), Q(1.2,2.728) $$

In Exercises 37-42, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. $$ y=x, \quad y=x^{2} ; \text { the line } y=2 $$

Find the centroid of the region under the graph of \(y=\sin \pi x\) on the interval \([0,1]\). Find the exact values of \(\bar{x}\) and \(\bar{y}\).

Find the volume of a frustum of a right circular cone with height \(h\), lower base radius \(R\), and upper radius \(r\).

Turbocharged Engine Versus Standard Engine In tests conducted by Auto Test Magazine on two identical models of the Phoenix Elite, one equipped with a standard engine and the other with a turbocharger, it was found that the acceleration of the former (in \(\mathrm{ft} / \mathrm{sec}^{2}\) ) is given by $$ a=f(t)=4+0.8 t \quad 0 \leq t \leq 12 $$ \(t\) sec after starting from rest at full throttle, whereas the acceleration of the latter (in \(\mathrm{ft} / \mathrm{sec}^{2}\) ) is given by $$ a=g(t)=4+1.2 t+0.03 t^{2} \quad 0 \leq t \leq 12 $$ How much faster is the turbocharged model moving than the model with the standard engine at the end of a \(10-\mathrm{sec}\) test run at full throttle?

A solid has a circular base of radius 2 , and its parallel cross sections perpendicular to its base are isosceles right triangles oriented so that the endpoints of the hypotenuse of a triangle lie on the circle. Find the volume of the solid.