• Left Riemann Sum (LRAM): $\sum_{i=1}^n f(x_{i-1})\Delta x$
• Right Riemann Sum (RRAM): $\sum_{i=1}^n f(x_i)\Delta x$
• Midpoint Riemann Sum (MRAM): $\sum_{i=1}^n f(\bar{x}_i)\Delta x$ where $\bar{x}_i = \frac{x_{i-1}+x_i}{2}$

$\int_a^b f(x)dx \approx \frac{\Delta x}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]$

More accurate than Riemann sums for smooth functions.

$\int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x$

where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is any point in ith subinterval.

Geometric interpretation: Signed area between f(x) and x-axis from x=a to x=b.

Fundamental Theorem of Calculus, Part 1 (FTC First Part)

$\frac{d}{dx}\left[\int_a^x f(t)dt\right] = f(x)$

Accumulation function: $F(x) = \int_a^x f(t)dt$

Fundamental Theorem of Calculus, Part 2 (FTC Second Part)

$\int_a^b f(x)dx = F(b) - F(a)$

where F is any antiderivative of f.

u-substitution: Let u = g(x), then du = g'(x)dx: $\int f(g(x))g'(x)dx = \int f(u)du$

For definite integrals, change limits: $\int_{x=a}^{x=b} f(g(x))g'(x)dx = \int_{u=g(a)}^{u=g(b)} f(u)du$

Used for rational functions where numerator degree ≥ denominator degree:

1. Polynomial long division
2. Complete the square for quadratic expressions

Improper integrals: $\int_a^\infty f(x)dx = \lim_{t \to \infty}\int_a^t f(x)dx$

$\int_a^b f(x)dx = \lim_{t \to b^-}\int_a^t f(x)dx$ (if f undefined at b)

Convergence: If limit exists and is finite -> converges; otherwise -> diverges

The Trapezoidal Rule Formula for Approximating the Area Under a Curve

$\int_a^b f(x)dx \approx \frac{\Delta x}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)]$

where $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$.