$\frac{a+b}{2} cdot h$, where a and b are the lengths of the parallel sides and h is the height.

$\Delta x = \frac{b-a}{n}$, where a and b are the interval endpoints and n is the number of subintervals.

$L_n = \Delta x [f(x_0) + f(x_1) + ... + f(x_{n-1})]$

$R_n = \Delta x [f(x_1) + f(x_2) + ... + f(x_n)]$

$T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]$

$M_n = \Delta x [f(\frac{x_0+x_1}{2}) + f(\frac{x_1+x_2}{2}) + ... + f(\frac{x_{n-1}+x_n}{2})]$

Left: Uses left endpoint for height, underestimates increasing functions, overestimates decreasing functions. Right: Uses right endpoint for height, overestimates increasing functions, underestimates decreasing functions.

Trapezoidal: Uses trapezoids, underestimates concave down, overestimates concave up. Midpoint: Uses midpoint height, underestimates concave up, overestimates concave down.

Left/Right: Easier to compute, less accurate. Trapezoidal/Midpoint: More complex, generally more accurate.

Approximation of the area under a curve using rectangles or trapezoids.

A Riemann sum where the height of each rectangle is determined by the function's value at the left endpoint of the subinterval.

A Riemann sum where the height of each rectangle is determined by the function's value at the right endpoint of the subinterval.

A Riemann sum where the height of each rectangle is determined by the function's value at the midpoint of the subinterval.

Approximation of the area under a curve using trapezoids instead of rectangles.

The base lengths formed by dividing up the total interval.

Left Riemann sum with n subdivisions.

Right Riemann sum with n subdivisions.

Trapezoidal Riemann sum with n subdivisions.

Midpoint Riemann sum with n subdivisions.