AP Calculus AB Unit 8: Applications of Integration
Applications of Integration
Average Value, Area, and Volume
1. Average Value of a Function
Finding the "average height" of a function f (x) over an interval [a, b].
Key Formula
Z b
1
favg = f (x) dx
b−a a
Interpretation: The height of a rectangle with width (b−a) that has the same area as the integral
of f (x).
2. Motion and Net Change
Connecting derivatives back to total accumulation.
• Displacement (Net Change in Position):
Z t2
v(t) dt = s(t2 ) − s(t1 )
t1
• Total Distance Traveled: (Must use absolute value of velocity!)
Z t2
|v(t)| dt
t1
• Current Position: Z t
s(t) = s(0) + v(x) dx
0
3. Area Between Two Curves
Finding the area of a region bounded by f (x) and g(x).
1
Applications of Integration
Average Value, Area, and Volume
1. Average Value of a Function
Finding the "average height" of a function f (x) over an interval [a, b].
Key Formula
Z b
1
favg = f (x) dx
b−a a
Interpretation: The height of a rectangle with width (b−a) that has the same area as the integral
of f (x).
2. Motion and Net Change
Connecting derivatives back to total accumulation.
• Displacement (Net Change in Position):
Z t2
v(t) dt = s(t2 ) − s(t1 )
t1
• Total Distance Traveled: (Must use absolute value of velocity!)
Z t2
|v(t)| dt
t1
• Current Position: Z t
s(t) = s(0) + v(x) dx
0
3. Area Between Two Curves
Finding the area of a region bounded by f (x) and g(x).
1
