91Ó°ÊÓ

For the region bounded by the graphs of the equations, find: (a) the volume of the solid formed by revolving the region about the \(x\) -axis and (b) the centroid of the region. $$ y=\sin x, y=0, x=0, x=\pi $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{0}^{\infty} f(x) d x\) converges

Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for \(k=1,0.1\), and 0.01 . Then evaluate the limit \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\).

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\cos a t $$

Let \(\int_{-\infty}^{\infty} f(x) d x\) be convergent and let \(a\) and \(b\) be real numbers where \(a \neq b\). Show that \(\int_{-\infty}^{a} f(x) d x+\int_{a}^{\infty} f(x) d x=\int_{-\infty}^{b} f(x) d x+\int_{b}^{\infty} f(x) d x\)