91Ó°ÊÓ

Find the surface area of the solid generated by revolving the region bounded by the graphs of \(y=x^{2}, y=0, x=0,\) and \(x=\sqrt{2} \quad\) about the \(x\) -axis.

The region bounded by the graph of \(f(x)=\frac{1}{1+x^{2}}\) and the \(x\) -axis between \(x=0\) and \(x=1\) is revolved about the \(x\) -axis. Find the volume of the solid that is generated.

Solve the initial-value problem for \(y\) as a function of \(x\). $$ \left(x^{2}+36\right) \frac{d y}{d x}=1, y(6)=0 $$

Solve the initial-value problem for \(y\) as a function of \(x\). $$ \left(64-x^{2}\right) \frac{d y}{d x}=1, y(0)=3 $$

Find the area bounded by \(y=\frac{2}{\sqrt{64-4 x^{2}}}, x=0, y=0,\) and \(x=2 .\)

An oil storage tank can be described as the volume generated by revolving the area bounded by \(y=\frac{16}{\sqrt{64+x^{2}}}, x=0, y=0, x=2\) about the \(x\) -axis. Find the volume of the tank (in cubic meters).

During each cycle, the velocity \(v\) (in feet per second) of a robotic welding device is given by \(v=2 t-\frac{14}{4+t^{2}}\), where \(t\) is time in seconds. Find the expression for the displacement \(s\) (in feet) as a function of \(t\) if \(s=0\) when \(t=0\)

Find the length of the curve \(y=\sqrt{16-x^{2}}\) between \(x=0\) and \(x=2\)

Express the rational function as a sum or difference of two simpler rational expressions. $$ \frac{1}{(x-3)(x-2)} $$

Express the rational function as a sum or difference of two simpler rational expressions. $$ \frac{x^{2}+1}{x(x+1)(x+2)} $$