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Chapter 5: Problem 1
Solve the differential equation. $$ \frac{d y}{d x}=x+2 $$
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The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(\sec ^{2} x-\cos x\right) d x $$
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.)
In Exercises \(57-60\), use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle \(x^{2}+(y-3)^{2}=4\) about the \(x\) -axis.
Find the area of the region by integrating (a) with respect to \(x\) and (b) with respect to \(y\). $$ \begin{array}{l} y=x^{2} \\ y=6-x \end{array} $$
(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. $$ y=\sqrt{1+x^{3}}, \quad y=\frac{1}{2} x+2, \quad x=0 $$
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