Science:Math Exam Resources/Courses/MATH101/April 2006/Question 02 (d)
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 (a) • Q4 (b) • Q5 • Q6 • Q7 •
Question 02 (d) 

FullSolution Problems. Justify your answers and show all your work. Simplification of answers is not required. Express the volume of the solid obtained by rotating the bounded region that lies between the curves and about the vertical line . Do not evaluate this integral. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

First, try drawing a picture, complete with intersection points. 
Hint 2 

Then use the method of cylindrical shells. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. First, we draw a picture. Setting the line and parabola equal to each other, we see that the points of intersection are at and so x=1 or x=4. This gives the following picture. The white area is what we want to rotate. Let x be a point between 1 and 4. Then the radius of a shell is r = 5x. The height of a shell is . Thus, when we rotate about x=5, the height will move a distance of and hence, the volume of revolution is 