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.

1. Determine $\Delta x$. 2. Identify the x-values for the chosen method (left, right, midpoint). 3. Calculate the function's value at each x-value. 4. Multiply each function value by $\Delta x$ and sum the results.

Determine if the function is increasing/decreasing (for left/right sums) or concave up/down (for midpoint/trapezoidal sums) on the interval.

1. Calculate $\Delta x$. 2. Find the y-values at each endpoint. 3. Apply the trapezoidal rule formula: $\frac{\Delta x}{2} [f(x_0) + 2f(x_1) + ... + f(x_n)]$.

1. Calculate $\Delta x$. 2. Find the midpoints of each subinterval. 3. Evaluate the function at each midpoint. 4. Multiply each function value by $\Delta x$ and sum the results.

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.