Unit 8 · Applications of Integration
● Core concept · ○ Supporting concept
8.1 Finding the Average Value of a Function on an Interval
Average value of a function ● (core concept) — The average of f over [a, b]: (1/(b − a))·∫ₐᵇ f(x) dx.
8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
Displacement ● (core concept) — The net change in position over [a, b]: ∫ₐᵇ v(t) dt; it can be negative.
Total distance traveled ● (core concept) — The full path length over [a, b]: ∫ₐᵇ |v(t)| dt; unlike displacement, it is never negative.
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
Net change ● (core concept) — The integral of a rate of change over an interval gives the net change: ∫ₐᵇ f'(x) dx = f(b) − f(a).
8.4 Finding the Area Between Curves Expressed as Functions of x
Area between curves (functions of x) ● (core concept) — For f ≥ g on [a, b], the area between the curves is ∫ₐᵇ [f(x) − g(x)] dx.
8.5 Finding the Area Between Curves Expressed as Functions of y
Area between curves (functions of y) ● (core concept) — For x = f(y) to the right of x = g(y) on [c, d], the area is ∫_c^d [f(y) − g(y)] dy.
8.7 Volumes with Cross Sections: Squares and Rectangles
Volume with cross sections ● (core concept) — If cross sections perpendicular to the x-axis have area A(x), the volume is ∫ₐᵇ A(x) dx (e.g., squares give A = s², rectangles A = l·w).
8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
Disc method ● (core concept) — Volume of a solid of revolution: V = π·∫ₐᵇ [R(x)]² dx, where R(x) is the radius from the curve to the axis.
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
Washer method ● (core concept) — Volume of a solid of revolution with a hole: V = π·∫ₐᵇ ([R(x)]² − [r(x)]²) dx, where R is the outer radius and r the inner radius.