Math 141H.1, Honors Calculus II

Bonus Problems 1

Stephen G. Simpson February 1, 2002

Please work alone. You may use a calculator or computer algebra system if you wish, but please give exact solutions and show all the steps needed to obtain your solutions by hand.

1. If f(x) = (10x)^{log_{10} x}, find the derivative f′(x).

2. Find the centroid of the region in the xy-plane described by

0 ≤ x ≤ \frac{π}{2} , 0 ≤ y ≤ sin x .

3. Let R be the region in the xy-plane described by

0 ≤ x ≤ 1 , 0 ≤ y ≤ \frac{1}{x^{4} + 2x^{2} + 1} .

Find the volume of the solid of revolution obtained by revolving R about the x-axis.

4. Evaluate the improper integral

∫_{0}^{∞} \frac{dx}{(x + 1)(x^{3} + 1)} .

5. Find

lim_{n→∞} \frac{ln(1 + 1/n) + ln(1 + 2/n) + ⋯ + ln(1 + n/n)}{n}

and justify your answer.