1. Clicker Question 1 What is the area enclosed by f(x) = 3x – x 2 and g(x) = x ? – A. 2/3 – B. 2 – C. 9/2 – D. 4/3 – E. 3

2. Clicker Question 2 What is the area enclosed by the y-axis, the line y = 2, and the curve y = 2 x ? – A. 2 – ln(2) – B. 2 – 1/ln(2) – C. 1 – D. 2 – E. ln(2) + 1

3. Finding Volumes Using the Integral (1/31/14) If we can see how to express the volume of a 3-dimensional figure by adding up little cross-sectional volumes, then we can use the integral to get the answer. This general technique does not (as we shall see) apply only to areas and volumes. It will apply in various applications to physics, probability, etc.

4. The general procedure Form cross-sections of your figure perpendicular to the x-axis, so that each cross-section has width  x). Describe the volumes of each cross-section in terms of x. Add up over all cross-sections. Let  x go to zero, obtaining a definite integral (  x gets replaced by dx). Compute the answer (exact or approximate).

5. A Simple Case: Volumes of Revolution If the cross-sections of a 3-dimensional object are circular in shape (i.e., are little disks), then we can obtain the volume by rotating whatever function describes the radius about the axis through the centers. Example: Find the volume generated if f (x) = e -x from x = 0 to x = 5 is revolved around the x-axis.

6. Clicker Question 3 What is the volume generated when the curve y = 1 / x between x = 1 and x = 3 is revolved around the x-axis? – A. ln(3) – B.  ln(3) – C. 2  / 3 – D. 2 / 3 – E. 26  / 9

7. Areas & Volumes of Famous Shapes First, how about a famous area. What is the area of a circle of radius r ? Let’s try using calculus to solve this problem. What is the volume of a sphere of radius r ? (Again, use calculus. This time we succeed!) What is the volume of a cone of radius r

8. Washers and Other Rotation Axes Even if the cross-sections are washers rather than disks, volumes are still easy to compute. Example: Find the volume generated when the area between y =  x and y = x/2 is revolved around the x-axis. What if the above area is rotated about the line y = -2?

9. Assignment for Monday Use calculus to derive the formula for the volume of a cone of radius r and height h. Read Section 6.2. Do Exercises 1, 3, 5, 7, 11, and 14 in that section.