Average Value Formula: (1/(b-a))∫abf(x)dx\int_a^b f(x)dx∫ab​f(x)dx

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

s(t) = ∫v(t)dt\int v(t)dt∫v(t)dt

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

v(t) = ∫a(t)dt\int a(t)dt∫a(t)dt

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

• Net change: $\int_a^b v(t)dt = s(b) - s(a)$ (signed)
• Total distance: $\int_a^b |v(t)|dt$ (always positive)

Interpretation of ∫∣v(t)∣dt\int |v(t)|dt∫∣v(t)∣dt

Gives total distance traveled, regardless of direction.

Area = ∫ab\int_a^b∫ab​ (top - bottom) dx

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

where f(x) ≥ g(x) on [a, b].

Area = ∫cd\int_c^d∫cd​ (right - left) dy

$A = \int_c^d [h(y) - k(y)]dy$

Divide at intersection points, apply formula on each subinterval.

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

Volume = ∫ab\int_a^b∫ab​ A(x) dx, where A(x) is area of cross-section

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

For square cross-sections: $A(x) = [s(x)]^2$ where s(x) is side length.

• Triangle: $A = \frac{1}{2}bh$
• Semicircle: $A = \frac{1}{2}\pi r^2$

Disc Method: π∫ab[R(x)]2dx\pi \int_a^b [R(x)]^2 dxπ∫ab​[R(x)]2dx or π∫cd[R(y)]2dy\pi \int_c^d [R(y)]^2 dyπ∫cd​[R(y)]2dy

$V = \pi \int_a^b [R(x)]^2 dx$

R(x) = radius (distance from curve to axis of rotation).

When rotating around line other than x- or y-axis, adjust radius accordingly.

Washer Method: π∫ab([R(x)]2−[r(x)]2)dx\pi \int_a^b ([R(x)]^2 - [r(x)]^2) dxπ∫ab​([R(x)]2−[r(x)]2)dx

$V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2)dx$

R(x) = outer radius, r(x) = inner radius.

Account for distance from rotation axis.