1. Chapter 7 AP Calculus BC

2. 7.1 Integral as Net Change Models a particle moving along the x- axis for t from 0 to 5. What is its initial velocity? When does it stop? v(5) = ??? Given s(0) = 9, find its position at t = 1 and t =5…. Total Distance traveled: you need…… This problem you need s(0), s(1.25), and s(5) On calculator, use absolute value. Work = F d Hooke’s Law (springs) – F = kx k = force constant for that spring x = distance you want to stretch or compress Example 7 p. 385

3. 7.2 Areas in the Plane Area between the curves – If f and g are cts. Functions with f(x) >g(x) throughout the interval [a,b] then the area between the curves is: Top - Bottom Examples: Integrating with respect to y (Right – Left) Choosing: (Both ways)

4. 7.3 Volumes (Cross Sections) Slicing a Cross – Section – Method: 1.Sketch the solid and a typical cross section (A(x)) 2.Find a formula for A(x) 3.Find the Limits of Integration 4.Integrate A(x) to find the Volume. Examples: Walk through #2 p. 406 Area Formulas:

5. 7.3 Volumes - Disks/Washers Two Methods for rotating around an axis, thus creating a volume. Question: Is the shaded region flat against the rotation axis? YES – Disk Method NO – Washer Method Examples: rotate around x-axis If you switch the rotation axis to something other than the axes (x or y). That number must included in all Radii.

6. 7.3 Cylindrical Shell Method In the shell method: If you rotate around the x-axis – you use y-values. If you rotate around the y-axis – you use x-values. Formulas : Around y – axis Around x - axis If you change the rotational axis, ONLY the radius changes!!!!!!!!!! Problem # 33 in book………

7. 7.4 Lengths of Curves Using a Riemann sum derive the length of curve formula. p.412-13……. Arc Length – Length of a Smooth Curve: If a smooth curve begins at (a,c) and ends at (b,d), then the length of the curve is : or Examples:

8. 7.5 Science Applications Work is Force(in the direction of motion) times displacement. Hooke’s Law: Force to stretch or compress a spring x – units, from its natural length is a constant times x. It takes a force of 10N to stretch a spring 2 m beyond its natural length. How much work is done in stretching the spring 6 m from its natural length? A leaky bucket weighs 22 N empty. It is lifted from the ground at a constant rate to a pt. 20 m above the ground by a rope weighing.4N/m. The bucket starts with 70 N of water but it leaks at a constant rate and just finishes draining as the bucket reaches the top. Find the Amount of Work done.

9. Leaky Bucket Solution: Lifting the bucket alone: Lifting the water alone: Lifting the rope alone: All together: 440 Nm 700 Nm 80 Nm 440 + 700 + 80 = 1220 Nm