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.

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.

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

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

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

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

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