91Ó°ÊÓ

What is "improper" about a definite integral over an interval on which the integrand has an infinite discontinuity? Explain why Definition 5.5.1 for \(\int_{a}^{b} f(x) d x\) fails if the graph of \(f\) has a vertical asymptote at \(x=a .\) Discuss a strategy for assigning a value to \(\int_{a}^{b} f(x) d x\) in this circumstance.

Use any method to find the area of the region enclosed by the curves. $$y=\frac{1}{25-16 x^{2}}, y=0, x=0, x=1$$

Use any method to find the area of the region enclosed by the curves. $$y=\sqrt{x} \ln x, y=0, x=4$$

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\) -axis. $$y=\cos x, \quad y=0, \quad x=0, \quad x=\pi / 2$$

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\) -axis. $$y=\sqrt{x-4}, \quad y=0, \quad x=8$$

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\) -axis. $$y=e^{-x}, y=0, x=0, x=3$$

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\) -axis. $$y=\ln x, \quad y=0, \quad x=5$$

Use any method to find the arc length of the curve. $$y=3 \ln x, 1 \leq x \leq 3$$

Use any method to find the area of the surface generated by revolving the curve about the \(x\) -axis. $$y=\sin x, \quad 0 \leq x \leq \pi$$

Use any method to find the area of the surface generated by revolving the curve about the \(x\) -axis. $$y=1 / x, \quad 1 \leq x \leq 4$$