91Ó°ÊÓ

\(\mathrm{A}\) lune is a crescent-shaped region bounded by the arcs of two circles. Let \(C_{1}\) be a circle of radius 4 centered at the origin. Let \(C_{2}\) be a circle of radius 3 centered at the point (2,0) Find the area of the lune (shaded in the figure) that lies inside \(C_{1}\) and outside \(C_{2}\)

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int p^{2} e^{-3 p} d p$$

The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \frac{d x}{e^{x}+e^{2 x}}$$

Use integration by parts to derive the following formulas for real numbers \(a\) and \(b\) $$\begin{aligned} &\int e^{a x} \sin b x d x=\frac{e^{a x}(a \sin b x-b \cos b x)}{a^{2}+b^{2}}+C\\\ &\int e^{a x} \cos b x d x=\frac{e^{a x}(a \cos b x+b \sin b x)}{a^{2}+b^{2}}+C \end{aligned}$$

Integrals of the form \(\int \sin m x \cos n x d x\) Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}(\cos ((m-n) x)-\cos ((m+n) x))\\\&\sin m x \cos n x=\frac{1}{2}(\sin ((m-n) x)+\sin ((m+n) x))\\\&\cos m x \cos n x=\frac{1}{2}(\cos ((m-n) x)+\cos ((m+n) x))\end{aligned}$$ $$\int \sin 5 x \sin 7 x d x$$

Use integration by parts to evaluate the following integrals. $$\int_{1}^{\infty} \frac{\ln x}{x^{2}} d x$$

Graph the integrands and then evaluate and compare the values of \(\int_{0}^{\infty} x e^{-x^{2}} d x\) and \(\int_{0}^{\infty} x^{2} e^{-x^{2}} d x\).

Integrals of the form \(\int \sin m x \cos n x d x\) Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}(\cos ((m-n) x)-\cos ((m+n) x))\\\&\sin m x \cos n x=\frac{1}{2}(\sin ((m-n) x)+\sin ((m+n) x))\\\&\cos m x \cos n x=\frac{1}{2}(\cos ((m-n) x)+\cos ((m+n) x))\end{aligned}$$ $$\int \sin 3 x \sin 2 x d x$$

Suppose a mass on a spring that is slowed by friction has the position function \(s(t)=e^{-t} \sin t\) a. Graph the position function. At what times does the oscillator pass through the position \(s=0 ?\) b. Find the average value of the position on the interval \([0, \pi]\) c. Generalize part (b) and find the average value of the position on the interval \([n \pi,(n+1) \pi],\) for \(n=0,1,2, \ldots\) d. Let \(a_{n}\) be the absolute value of the average position on the intervals \([n \pi,(n+1) \pi],\) for \(n=0,1,2, \ldots .\) Describe the pattern in the numbers \(a_{0}, a_{1}, a_{2}, \ldots\)

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int \tan ^{4} 3 y d y$$