91Ó°ÊÓ

A square plate of side \(a\) feet is dipped in a liquid of weight density \(\rho\) Ib/ft \(^{3} .\) Find the fluid force on the plate if a vertex is at the surface and a diagonal is perpendicular to the surface.

Determine whether the statement is true or false. Explain your answer. A force of 10 lb in the direction of motion of an object that moves \(5 \mathrm{ft}\) in \(2 \mathrm{s}\) does six times the work of a force of 10 ib in the direction of motion of an object that moves \(5 \mathrm{ft}\) in \(12 \mathrm{s}\)

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the \(x\) -axis. $$y=\sqrt{25-x^{2}}, y=3$$

Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the \(y\) -axis. $$y=e^{x^{2}}, x=1, x=\sqrt{3}, y=0$$

Sketch the region enclosed by the curves and find its area. $$x^{2}=y, x=y-2$$

Find the volume of the solid that results when the region enclosed by the given curves is revolved about the \(x\) -axis. $$y=9-x^{2}, y=0$$

Find the centroid of the region. The triangle with vertices \((0,0),(1,1),\) and (2,0).

Find \(d y / d x\). $$y=\ln (\tanh 2 x)$$

Determine whether the statement is true or false. Explain your answer. It follows from Hooke's law that in order to double the distance a spring is stretched beyond its natural length, four times as much work is required.

Use a CAS to find the exact area of the surface generated by revolving the curve about the stated axis. $$x=\sqrt{16-y}, 0 \leq y \leq 15 ; \text { y-axis }$$