91Ó°ÊÓ

It took 1800 J of work to stretch a spring from its natural length of 2 m to a length of 5 m. Find the spring’s force constant.

Find the center of mass of a thin plate of constant density \(\delta\) covering the given region.s The region bounded by the parabola \(y=x-x^{2}\) and the line \(y=-x\)

a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. \(x^{1 / 2}+y^{1 / 2}=3 \quad\) from (4,1) to (1,4)\(; \quad x\) -axis

Find the center of mass of a thin plate of constant density \(\delta\) covering the given region.s The region bounded by the parabola \(x=y^{2}-y\) and the line \(y=x\)

Find the volumes of the solids. The base of a solid is the region bounded by the graphs of \(y=3 x, y=6,\) and \(x=0 .\) The cross-sections perpendicular to the \(x\) -axis are a. rectangles of height \(10 .\) b. rectangles of perimeter \(20 .\)

Lifting a rope A mountain climber is about to haul up a \(50-\mathrm{m}\) length of hanging rope. How much work will it take if the rope weighs \(0.624 \mathrm{N} / \mathrm{m} ?\)

A bag of sand originally weighing 144 lb was lifted at a constant rate. As it rose, sand also leaked out at a constant rate. The sand was half gone by the time the bag had been lifted to 18 ft. How much work was done lifting the sand this far? (Neglect the weight of the bag and lifting equipment.)

Find the lateral (side) surface area of the cone generated by revolving the line segment \(y=x / 2,0 \leq x \leq 4,\) about the \(x\) -axis. Check your answer with the geometry formula Lateral surface area \(=\frac{1}{2} \times\) base circumference \(\times\) slant height.

Find the center of mass of a thin plate of constant density \(\delta\) covering the given region.The region cut from the first quadrant by the circle \(x^{2}+y^{2}=9\) b. The region bounded by the \(x\) -axis and the semicircle \(y=\sqrt{9-x^{2}}\) Compare your answer in part (b) with the answer in part (a).

Find the surface area of the cone frustum generated by revolving the line segment \(y=(x / 2)+(1 / 2), 1 \leq x \leq 3,\) about the y-axis. Check your result with the geometry formula Frustum surface area \(=\pi\left(r_{1}+r_{2}\right) \times\) slant height.