1. Calculus 7.1: Linear Motion
A. DEF: Displacement—a change in position.
< if s(t) is a body’s position at time t, the displacement over [t1,t2] is s(t2) – s(t1) >
B. Displacement vs. Total Distance
1. Displacement =
2. Total Distance =
3. New position = initial position + displacement

2. Calculus 7.2: Net Change Theorem
A. The integral of a rate of change is the net change (or total accumulation) from a to b:

3. Calc 7.3: Areas Between Curves
A. The Area of the region bounded by the curves y=f(x), y=g(x), and the lines x=a and x=b (f and g continuous and for all x in [a,b]) is:

4. B. Examples (find area bounded by…)
1. y = x2 + 1, y = x, x = 0, x = 1
2. y = x2 , y = 2x – x2
3. y = x4 – x,
4. (#50) y = sin x, y = ex, x = 0, x =

5. Calculus 7.4: Area Between Curves (dy)
A. Integrating with respect to y
<if the curves can be written as x=f(y) and x=g(y), y=c, y=d where f(y) > g(y) on [c,d]>
Ex 1: (find the area of the region bounded by…)
a) y = x – 1 , y2 = 2x + 6
b) y = x2 , y2 = x
c) y = x3 , x = y2 – 2

6. Finding Areas of Multiple Sub-regions
<find the sum of the sub-regions
A1 + A2 + A3 + etc.>
Ex 2: (find the area of the region bounded by…)
a) y = sin x, y = cos x, x = 0, x =
b) x-axis, y = x – 2,
c) (#47) find the value of c such that the area enclosed by y = x2 – c2 and y = c2 – x2 is 576

7. Calculus 7.5: Volumes of Solids
A. Calculating Volume:
1. For any cylinder, V = Ah
<A = Area of base, h = height>
2. For any non-cylinder, find volume by “cutting” the solid into thinly-sliced cylinders:
a) Let A(x) = Area of cross-section
b) Let = width (thickness) of each slice
c) Now

8. Definition of Volume:
Let S be a solid that lies between x=a and x=b. If the cross-sectional Area of S in the plane P, through x and perpendicular to the x-axis, is A(x) <a continuous function> then the volume of S is:

9. Calculus 7.6: Volumes of Solids of Revolution
A. Solids obtained by rotating about x = a or y = b
1. Find the volume of the solid obtained by rotating the region bounded by y = x and y = x2 , about the line y = 2
2. Find the volume of the solid by rotating the same region about the line x = -1

10. Solids of Revolution
1. If cross-section is a disk use
2. If cross-section is a washer use

11. Calculus 7.7: Volumes of Other Solids
A. Solids with Known Cross-Sections
1. Solid with base enclosed by
y = 2 sin x and the x-axis, cross-sections are semicircles (see p. 403)
2. Solid with circular base (r = 1) and cross-sections are equilateral triangles (see Stewart p. 388)

12. Calculus Unit 7 Test
Grademaster #1-20 (Name, Date, Subject, Period, Test Copy #)
Do Not Write on Test! Show All Work on Scratch Paper!
Label BONUS QUESTIONS Clearly on Notebook Paper. (If you have time)
Find Something QUIET To Do When Finished!