91Ó°ÊÓ

Finding the Area of a Region In Exercises \(15 - 28\) sketch the region bounded by the graphs of the equations and find the area of the region. $$y = x ^ { 2 } - 1 , \quad y = - x + 2 , \quad x = 0 , \quad x = 1$$

In Exercises 13-22, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. $$y=x^{3}, x=0, y=8$$

In Exercises 13-22, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. $$y=3-x, y=0, x=6$$

Finding the Volume of a Solid In Exercises\(21 - 24\) , find the volume of the solid generated by revolving the region bounded by the graphs of the Nequations about the line \(x = 5 .\) $$y = x , \quad y = 0 , \quad y = 4 , \quad x = 5$$

Center of Mass of a Planar Lamina In Exercises \(15-28,\) find \(M_{x}, M_{y},\) and \((\overline{x}, \overline{y})\) for the lamina of uniform density \(\rho\) bounded by the graphs of the equations. $$y=x^{2 / 3}, y=0, x=8$$

In Exercises 23-26, use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given line. $$y=\frac{1}{3} x^{3}, \quad y=6 x-x^{2}, \text { about the line } x=3$$

Finding the Volume of a Solid In Exercises \(25 - 32 ,\) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$y = x \sqrt { 4 - x ^ { 2 } } , \quad y = 0$$

Finding the Volume of a Solid In Exercises \(25 - 32 ,\) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$y = \frac { 2 } { x + 1 } , \quad y = 0 , \quad x = 0 , \quad x = 6$$

Comparing Methods In Exercises 29 and 30,find the area of the region by integrating (a) with respect to x and (b) with respect to y. (c) Compare your results. Which method is simpler? In general, will this method always be simpler than the other one? Why or why not? $$\begin{array} { l } { y = x ^ { 2 } } \\ { y = 6 - x } \end{array}$$

Finding Arc Length In Exercises \(21-30\) , (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) write a definite integral that represents the that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the arc length.arc length of the curve over the indicated interval and observe $$x=\sqrt{36-y^{2}}, \quad 0 \leq y \leq 3$$