91Ó°ÊÓ

The given expression is the limit of a Riemann sum of a function fon \([a, b] .\) Write this expression as a definite integral on \([a, b]\). \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n} c_{k}\left(\cos c_{k}\right) \Delta x, \quad\left[0, \frac{\pi}{2}\right]\)

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with n subintervals. $$ \int_{0}^{1} \frac{d x}{x+1} ; \quad n=8 $$

Evaluate the integral. $$ \int_{\pi / 4}^{\pi / 3} \frac{d x}{\sin ^{2} x \cos ^{2} x} $$

Evaluate the limit after first finding the sum (as a function of \(n\) ) using the summation formulas. $$ \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(1+\frac{2 k-1}{2 n}\right)\left(\frac{1}{n}\right) $$

Determine whether the Dirichlet function $$ f(x)=\left\\{\begin{array}{ll} 1 & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational } \end{array}\right. $$ is integrable on the interval \([0,1]\). Explain.

A horizontal, uniform beam of length \(L\), supported at its ends, bends under its own weight, \(w\) per unit length. The elastic curve of the beam (the shape that it assumes) has equation \(y=f(x)\) satisfying $$ E I y^{\prime \prime}=\frac{w x^{2}}{2}-\frac{w L x}{2} $$ where \(E\) and \(I\) are positive constants that depend on the material and the cross section of the beam. a. Find an equation of the elastic curve. Hint: \(y=0\) at \(x=0\) and at \(x=L\). b. Show that the maximum deflection of the beam occurs at \(x=L / 2\)

The water level (in feet) in Boston Harbor during a certain 24 -hr period is approximated by the formula $$H=4.8 \sin \left[\frac{\pi}{6}(t-10)\right]+7.6 \quad 0 \leq t \leq 24$$ where \(t=0\) corresponds to 12 A.M. What is the average water level in Boston Harbor over the 24 -hr period on that day?