91Ó°ÊÓ

In Exercises \(7-51\), find or evaluate the integral. $$ \int \frac{\sin x}{\cos ^{3} x+\cos ^{2} x} d x $$

Find the surface area of the ellipsoid formed by revolving the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b\), about the \(x\) -axis.

Use the Comparison Test to determine whether the integral is convergent or divergent by comparing it with the second integral. $$ \int_{1}^{\infty} \frac{\cos ^{2} x}{x^{2}} d x ; \quad \int_{1}^{\infty} \frac{1}{x^{2}} d x $$

Find or evaluate the integral. $$ \int \frac{1}{\csc x \cot ^{2} x} d x $$

\begin{array}{l} \text { Find the area of the region under the graph of } y=\frac{x e^{2}}{(1+x)^{2}} \\ \text { on the interval }[0,1] . \end{array}

Use the Comparison Test to determine whether the integral is convergent or divergent by comparing it with the second integral. $$ \int_{1}^{\infty} \frac{d x}{x+\sin ^{2} x} ; \quad \int_{1}^{\infty} \frac{1}{1+x} d x $$

In Exercises \(7-51\), find or evaluate the integral. $$ \int \frac{\sec ^{2} \theta}{\tan \theta(\tan \theta-1)} d \theta $$

Average Mass of an Electron According to the special theory of relativity, the mass \(m\) of a particle moving at a velocity \(v\) is given by $$ m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}} $$ where \(m_{0}\) is the mass of the body at rest and \(c=3 \times 10^{8}\) \(\mathrm{m} / \mathrm{sec}\) is the speed of light. If an electron is accelerated from a speed of \(v_{1} \mathrm{~m} / \mathrm{sec}\) to a speed of \(v_{2} \mathrm{~m} / \mathrm{sec}\), find an expression for the average mass of the electron between \(v=v_{1}\) and \(v=v_{2}\).

Find the force exerted by a liquid of constant weight density \(\delta\) on a vertical ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) whose center is sub- merged in the liquid to a depth \(h\), where \(h \geq b\).

Find the area of the surface generated by revolving the graph of \(y=x^{2}\) for \(0 \leq x \leq 1\) about the \(x\) -axis.