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Chapter 7

Chapter 7. AP Calculus BC. 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…….

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Chapter 7

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  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. Hooke’s Law (springs) – F = kx k = force constant for that spring x = distance you want to stretch or compress Work = F d 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: Sketch the solid and a typical cross section (A(x)) Find a formula for A(x) Find the Limits of Integration 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 If you switch the rotation axis to something other than the axes (x or y). That number must included in all Radii. Examples: rotate around x-axis

  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: 440 Nm Lifting the water alone: 700 Nm Lifting the rope alone: 80 Nm All together: 440 + 700 + 80 = 1220 Nm

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