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.

$\frac{a+b}{2} cdot h$, 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.

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

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

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

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

By dividing the area into rectangles or trapezoids and summing their areas. As the number of subdivisions increases, the approximation becomes more accurate.

When the function is increasing.

When the function is increasing.

When the function is decreasing.

When the function is decreasing.

When the function is concave down.

When the function is concave up.

When the function is concave up.

When the function is concave down.

They minimize error by either using the midpoint height or averaging left and right endpoint heights.