Chapter 5: Problem 3
Solve the differential equation. $$ y^{\prime}=\frac{5 x}{y} $$
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(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=x^{4}-2 x^{2}, \quad y=2 x^{2} $$
Fluid Force on a Circular Plate A circular plate of radius \(r\) feet is submerged vertically in a tank of fluid that weighs \(w\) pounds per cubic foot. The center of the circle is \(k(k>r)\) feet below the surface of the fluid. Show that the fluid force on the surface of the plate is \(F=w k\left(\pi r^{2}\right)\) (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.)
(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ g(x)=\frac{4 \ln x}{x}, \quad y=0, \quad x=5 $$
The region bounded by \(y=\sqrt{x}, y=0, x=0,\) and \(x=4\) is revolved about the \(x\) -axis. (a) Find the value of \(x\) in the interval [0,4] that divides the solid into two parts of equal volume. (b) Find the values of \(x\) in the interval [0,4] that divide the solid into three parts of equal volume.
A sphere of radius \(r\) is cut by a plane \(h(h
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