$f_{avg} = \frac{1}{b-a}\int_a^b f(x)dx$

s(t) = ∫v(t)dt

$s(t) = \int v(t)dt$

Position is the integral of velocity.

v(t) = ∫a(t)dt

$v(t) = \int a(t)dt$

Velocity is the integral of acceleration.

Displacement: $\int_{t_1}^{t_2} v(t)dt = s(t_2) - s(t_1)$ (can be negative)

Total distance: $\int_{t_1}^{t_2} |v(t)|dt$ (always positive)

• Net change: $\int_a^b v(t)dt$ (signed area)
• Total distance: $\int_a^b |v(t)|dt$ (actual distance traveled)

The Integral of a Rate of Change

The integral of a rate of change gives the total net change in the quantity.

Example:

• If $\frac{dQ}{dt}$ represents rate of flow, then $\int_{t_1}^{t_2} \frac{dQ}{dt}dt = Q(t_2) - Q(t_1)$
• Net change = final value - initial value

Area = ∫[a,b] (Top - Bottom) dx

$A = \int_a^b [f(x) - g(x)]dx$

where f(x) ≥ g(x) on [a, b].

Area = ∫[c,d] (Right - Left) dy

$A = \int_c^d [R(y) - L(y)]dy$

where R(y) ≥ L(y) on [c, d].

When curves intersect at multiple points:

1. Find all intersection points
2. Divide interval at intersection points
3. Apply area formula to each subinterval
4. Sum all areas

$A = \int_a^c |f(x) - g(x)|dx$

Takes absolute value to handle regions where curves cross.

Volume = ∫[a,b] A(x) dx, Where A(x) is Area of Cross-Section

For solids with known cross-sectional area A(x): $V = \int_a^b A(x)dx$

Square cross-sections: If side = s(x), then $A(x) = [s(x)]^2$ Rectangular cross-sections: If dimensions are w(x) and h(x), then $A(x) = w(x) \cdot h(x)$

Triangle: $A = \frac{1}{2}bh$ (base × height) Semicircle: $A = \frac{1}{2}\pi r^2$ (half of circle area)

For revolving region under f(x) from x=a to x=b about x-axis: $V = \pi \int_a^b [f(x)]^2 dx$

For region between f(x) (top) and g(x) (bottom): $V = \pi \int_a^b ([f(x)]^2 - [g(x)]^2)dx$

$V = \int_a^b A(x)dx$

where A(x) is area of cross-section perpendicular to x-axis.

Cross-section shapes:

• Square: side length from problem
• Rectangle: dimensions from problem
• Semicircle: $A = \frac{\pi}{8}[d(x)]^2$ where d(x) is diameter