Did you read my tips on how to study and learn Math 1700?  If not, here is a link to those important suggestions:

Did you read my Tips for Assignment 1?  If not, here is a link to those important suggestions:

These are tips for the second assignment in the Distance/Online Math 1700 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course.  These assignments are very good (and challenging) practice.  It is possible that you are doing the topics in a different order in the classroom lecture sections, so you may need to wait until later before tackling this assignment.

Here is a link to the actual assignment, in case you don't have it:

Note that you need to study Lesson 5 (Area Between 2 Curves), Lesson 6 (Volumes), Lesson 7 (Integrals of Trigonometric Functions), and Lesson 8 (Integration by Parts) from my Calculus 2 book to prepare for this assignment!  I think you should find this assignment fairly straightforward if you do thoroughly study and do all the Practise Problems I give you in these lessons.

Don't have my book?  You can download a free sample of my book and audio lectures containing Lesson 1 and 11:

You can try to find the points of intersection algebraically, but, the fact is, the best way to do things like that for transcendental functions (trigonometric functions, logarithmic functions, exponential functions, inverse trigonometric functions) is to just use trial and error.  Sub in x=0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, 5pi/6, and pi and just see which of those are points where the curves intersect.  This will also help you sketch the curve.

Obviously, you will have to solve the two volumes they describe to see which is bigger.

Quite similar to my Lesson 6, question 2.  Be sure to sketch the curve, including the vertical lines at x=1 and x=4, and the horizontal line at y=0, and shade the bounded region.

If you have studied Lessons 7 and 8, these should not prevent much difficulty.  I am confused by the hint in part (a) though.  Fact is there is no need for integration by parts here.  It is actually unnecessarily complicated to use that method.