91Ó°ÊÓ

\(1-8=\) Solve the differential equation. $$\frac{d y}{d x}=x y^{2}$$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. $$y=2-\frac{1}{2} x, y=0, x=1, x=2 ; \text { about the } x-axis$$

\(1-4=\) (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the \(x\) -axis and (ii) the \(y\) -axis. (b) Use the numerical integration capability of your calculator to evaluate the surface areas correct to four decimal places. $$y=\tan x, \quad 0 \leqslant x \leqslant \pi / 3$$

A variable force of 5\(x^{-2}\) pounds moves an object along a straight line when it is \(x\) feet from the origin. Calculate the work done in moving the object from \(x=1\) ft to \(x=10 \mathrm{ft.}\)

When a particle is located a distance \(x\) meters from the origin, a force of \(\cos (\pi x / 3)\) newtons acts on it. How much work is done in moving the particle from \(x=1\) to \(x=2 ?\) Interpret your answer by considering the work done from \(x=1\) to \(x=1.5\) and from \(x=1.5\) to \(x=2.\)

\(1-4=\) (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the \(x\) -axis and (ii) the \(y\) -axis. (b) Use the numerical integration capability of your calculator to evaluate the surface areas correct to four decimal places. $$y=x^{-2}, \quad 1 \leq x \leqslant 2$$

Use the arc length formula to find the length of the curve \(y=\sqrt{2-x^{2}}, 0 \leqslant x \leqslant 1 .\) Check your answer by noting that the curve is part of a circle.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. $$y=1-x^{2}, y=0 ; \quad \text { about the } x-axis$$

\(1-8=\) Solve the differential equation. $$\frac{d y}{d x}=x e^{-y}$$

\(1-8=\) Solve the differential equation. $$x y^{2} y^{\prime}=x+1$$