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Magazine Problem 56
Find the area of the region between \(y=x \sin x\) and \(y=x\) for \(0 \leq x \leq \pi / 2\).
Problem 57
(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{d x}{x-\sqrt[3]{x}} $$
Problem 57
Find the volume of the solid generated when the region between \(y=\sin x\) and \(y=0\) for \(0 \leq x \leq \pi\) is revolved about the \(y\) -axis.
Problem 58
Find the volume of the solid generated when the region enclosed between \(y=\cos x\) and \(y=0\) for \(0 \leq x \leq \pi / 2\) is revolved about the \(y\) -axis.
Problem 58
(a) Give a reasonable informal argument, based on areas, that explains why the integrals $$ \int_{0}^{+\infty} \sin x d x \text { and } \int_{0}^{+\infty} \cos x d x $$ diverge. (b) Show that \(\int_{0}^{+\infty} \frac{\cos \sqrt{x}}{\sqrt{x}} d x\) diverges.
Problem 58
(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{d x}{\sqrt{x}+\sqrt[3]{x}} $$
Problem 59
In electromagnetic theory, the magnetic potential at a point on the axis of a circular coil is given by $$ u=\frac{2 \pi N I r}{k} \int_{a}^{+\infty} \frac{d x}{\left(r^{2}+x^{2}\right)^{3 / 2}} $$ where \(N, I, r, k\), and \(a\) are constants. Find \(u\).
Problem 59
Find the arc length of the curve \(y=\ln (\cos x)\) over the interval \([0, \pi / 4]\).
Problem 59
(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{d x}{x\left(1-x^{1 / 4}\right)} $$
Problem 59
A particle moving along the \(x\) -axis has velocity function \(v(t)=t^{3} \sin t .\) How far does the particle travel from time \(t=0\) to \(t=\pi ?\)
