Approximating Areas

The area problem returns

Left, right, and midpoint sums

Refinement and convergence

Summation notation and shortcuts

A Riemann sum for f on [a, b] with n rectangles is fundamentally an approximation of what quantity?

For a monotone function on an interval, why does the midpoint rule typically beat both endpoint rules?

For an increasing function on [a, b], how do the left and right Riemann sums relate to the exact area?

For the left and right endpoint rules, doubling the number of rectangles roughly does what to the error?

The interval [2, 5] is partitioned into 6 equal subintervals. What is the width Delta x of each subinterval?

The definition of the definite integral as a limit of Riemann sums requires the limit to be the same regardless of the sample points x_i star chosen in each subinterval. Why is this independence important?

For f(x) = x on [0, 1] with n = 4 rectangles of width 1/4, the left endpoints are 0, 1/4, 1/2, 3/4. What is the left Riemann sum? Answer with a fraction like 3/8.

Each term of a Riemann sum has the form f(x_i star) times Delta x. What does this term represent geometrically? Answer in two words like strip width.

A strip spans [x_{i-1}, x_i]. The midpoint rule samples the function at the strip's center. What is that center, expressed with x_{i-1} and x_i? Answer like (a + b)/2.

The Riemann sums for x^2 on [0, 1] converge to a famous exact value as n grows. What is that value, as a fraction? Answer like 1/3.