91Ó°ÊÓ

If \(\bar{x}\) is the \(x\) -coordinate of the centroid of the region that lies under the graph of a continuous function \(f,\) where \(a \leqslant x \leqslant b\) show that $$ \int_{a}^{b}(c x+d) f(x) d x=(c \bar{x}+d) \int_{a}^{b} f(x) d x $$

A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. high by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation \(y=\sin (\pi x / 7)\) and find the width \(w\) of a flat metal sheet that is needed to make a 28 -inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)

Find the length of the curve $$ y=\int_{1}^{x} \sqrt{t^{3}-1} d t \quad 1 \leqslant x \leqslant 4 $$

The curves with equations \(x^{n}+y^{n}=1, n=4,6,8, \ldots,\) are called fat circles. Graph the curves with \(n=2,4,6,8,\) and 10 to see why. Set up an integral for the length \(L_{2 k}\) of the fat circle with \(n=2 k .\) Without attempting to evaluate this integral, state the value of \(\lim _{k \rightarrow \infty} L_{2 k} .\)

Use the Theorem of Pappus to ind the volume of the given solid. $$ \begin{array}{l}{\text { The solid obtained by rotating the triangle with vertices }} \\ {(2,3),(2,5), \text { and }(5,4) \text { about the } x \text { -axis }}\end{array} $$

Let \(\mathscr{R}\) be the region that lies between the curves $$ y=x^{m} \quad y=x^{n} \quad 0 \leqslant x \leqslant 1 $$ where \(m\) and \(n\) are integers with \(0 \leqslant n