# Calculus Worksheet-CSUN .

Instructions: For the following problems, you must show and explain all of your work in complete sentences. Collaboration is allowed with class- mates, but you must state for each problem who you worked with and what resources you used. Problem 1. a. We showed in class that dr C diverges, but that 1 su dr CP converges for any p > 1. Show both of these things by evaluating the integrals above directly. Make sure to explain where the assumption p > 1 is used. b. The previous part shows that f(x) = 1/x doesn’t go to zero fast enough as r → in order to have finite area under its graph, but multiplying the denominator by any small positive power of x (so, for example, multiplying by 20.01) is enough to make the area under the graph of the new function finite. It is interesting to wonder if multiplying by some other slowly growing function might be enough to do the trick. We know that In(1) o as * + 20. It follows that „In(a) + 0 faster than as x + 0. Nevertheless, show through direct evaluation that 1 dc x ln(2) diverges. Hint: you can use u-sub here. c. Now show that increasing the power of ln(x) by a little bit is enough to make the integral converge; show that 1 dr x ln(2)P converges for any p > 1. Hint: u-sub works here too, and make sure to explain where the assumption p > 1 was used. ſi zinc 1 2 UCSD 10B, SPRING 2020: HOMEWORK 4 Problem 2. a. Draw the region R enclosed by the curves y = Vix and y = x2. Make sure to label the intersection points on your graph. b. Find the area of the region R by integrating along the x-axis. Show all of your work, and draw a typical approximating rectangle. c. Find the volume of the solid obtained by rotating R about the z-axis. Show all of your work, and draw a typical approximating washer. d. Find the volume of the solid obtained by rotating R about the y-axis. Show all of your work, and draw a typical approximating washer.