Clicker Question 1 What is x2 (x3 + 4)20 dx ? A. (1/63)x3(x3 + 4)21 + C B. (1/63)(x3 + 4)21 + C C. 21x3(x3 + 4)21 + C D. 63x3(x3 + 4)21 + C E. 21(x3 + 4)21 + C

Clicker Question 2 What is the average value of f (x) = ex + 1 on the interval [0, 2]? A. e2 + 1 B. (e2 + 1) / 2 C. (e2 + 2) / 2 D. e2 + 2 E. (2e + 1) / 2

Finding Volumes Using the Integral (9/11/13) If we can see how to express volume of a 3-dimensional figure by adding up little cross-sectional volumes, then we can use the integral to get the answer. We’ve already seen this with areas (Calc I). 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.

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).

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.

Clicker Question 3 What volume is generated when the curve y = 1/x2 between x = 1 and x = 2 is revolved about the x- axis? A.  / 2 B. 1 / 2 C. 7 / 24 D. 7 / 24 E. -  /2

Volumes of Famous Shapes What is the volume of a sphere of radius r ? What is the volume of a cone of radius r and of height h?

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?

Assignment for Friday Read Section 6.2. Do Exercises 1, 3, 5, 7, 11, and 14 in that section.