Explain how Riemann sums approximate the area under a curve.
By dividing the area into rectangles or trapezoids and summing their areas. As the number of subdivisions increases, the approximation becomes more accurate.
When does a Left Riemann Sum underestimate the area?
When the function is increasing.
When does a Right Riemann Sum overestimate the area?
When the function is increasing.
When does a Left Riemann Sum overestimate the area?
When the function is decreasing.
When does a Right Riemann Sum underestimate the area?
When the function is decreasing.
When does a Trapezoidal Riemann Sum underestimate the area?
When the function is concave down.
When does a Trapezoidal Riemann Sum overestimate the area?
When the function is concave up.
When does a Midpoint Riemann Sum underestimate the area?
When the function is concave up.
When does a Midpoint Riemann Sum overestimate the area?
When the function is concave down.
How do midpoint and trapezoidal Riemann sums improve accuracy?
They minimize error by either using the midpoint height or averaging left and right endpoint heights.
What is the formula for the area of a trapezoid?
$\frac{a+b}{2} cdot h$, where a and b are the lengths of the parallel sides and h is the height.
How do you calculate the width of each subinterval in a Riemann sum?
$\Delta x = \frac{b-a}{n}$, where a and b are the interval endpoints and n is the number of subintervals.
Formula for Left Riemann Sum
$L_n = \Delta x [f(x_0) + f(x_1) + ... + f(x_{n-1})]$
Formula for Right Riemann Sum
$R_n = \Delta x [f(x_1) + f(x_2) + ... + f(x_n)]$
Formula for Trapezoidal Riemann Sum
$T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]$
Formula for Midpoint Riemann Sum
$M_n = \Delta x [f(\frac{x_0+x_1}{2}) + f(\frac{x_1+x_2}{2}) + ... + f(\frac{x_{n-1}+x_n}{2})]$
Steps to calculate a Riemann Sum given a function and interval.
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.
How to determine if a Riemann sum is an over or underestimate?
Determine if the function is increasing/decreasing (for left/right sums) or concave up/down (for midpoint/trapezoidal sums) on the interval.
Steps to calculate a Trapezoidal Sum.
1. Calculate $\Delta x$. 2. Find the y-values at each endpoint. 3. Apply the trapezoidal rule formula: $\frac{\Delta x}{2} [f(x_0) + 2f(x_1) + ... + f(x_n)]$.
Steps to calculate a Midpoint Sum.
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.
