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Chapter 5: Problem 39
If the portion of the line \(y=\frac{1}{2} x\) lying in the first quadrant is revolved about the \(x\) -axis, a cone is generated. Find the volume of the cone extending from \(x=0\) to \(x=6\).
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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. $$ f(x)=x^{4}-4 x^{2}, \quad g(x)=x^{2}-4 $$
(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places. $$ y=\sqrt{x} e^{x}, \quad y=0, x=0, x=1 $$
Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ y=-\frac{3}{8} x(x-8), y=10-\frac{1}{2} x, x=2, x=8 $$
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.)
Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. $$ y=x^{3}-2 x, \quad(-1,1) $$
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