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

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:

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:

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 =

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

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

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

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:

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

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

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)

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!