91Ó°ÊÓ

In Exercises \(11-18\) , find the exact length of the curve analytically by antidifferentiation. You will need to simplify the integrand algebraically before finding an antiderivative. $$x=\left(y^{3 / 3}\right)+1 /(4 y) \quad \text { from } y=1 \text { to } y=3$$ [Hint: \(1+(d x / d y)^{2}\) is a perfect square. \(]\)

In Exercises 11-20, find the volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis. $$y=\sqrt{9-x^{2}}, \quad y=0$$

In Exercises 11-20, find the volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis. $$y=x-x^{2}, \quad y=0$$

In Exercises \(11-18\) , find the exact length of the curve analytically by antidifferentiation. You will need to simplify the integrand algebraically before finding an antiderivative. $$x=\left(y^{4 / 4}\right)+1 /\left(8 y^{2}\right) \quad \text { from } y=1 \text { to } y=2$$ $$\left[\text {Hint:} 1+(d x / d y)^{2} \text { is a perfect square. }\right]$$

In Exercises \(11-18\) , find the exact length of the curve analytically by antidifferentiation. You will need to simplify the integrand algebraically before finding an antiderivative. $$x=\left(y^{3 / 6}\right)+1 /(2 y) \quad \text { from } y=1 \text { to } y=2$$ $$\left[\text {Hint} : 1+(d x / d y)^{2} \text { is a perfect square. }\right]$$

In Exercises \(15-34,\) find the area of the regions enclosed by the lines and curves. $$y=x^{2}-2$$ and $$y=2 \quad$$

In Exercises 11-20, find the volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis. $$y=x, \quad y=1, \quad x=0$$

In Exercises 11-20, find the volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis. $$y=2 x, \quad y=x, \quad x=1$$

In Exercises \(15-34,\) find the area of the regions enclosed by the lines and curves. $$y=2 x-x^{2} \quad$$ and $$\quad y=-3$$

In Exercises \(11-18\) , find the exact length of the curve analytically by antidifferentiation. You will need to simplify the integrand algebraically before finding an antiderivative. $$y=\left(x^{3 / 3}\right)+x^{2}+x+1 /(4 x+4), \quad 0 \leq x \leq 2$$