Integration and Accumulation of Change

Given the function $f(x) = x + 1$ on the interval $[0, 2]$, approximate the area under the curve using a left Riemann sum with $n=2$ subintervals. What is the value of $\Delta x$?

Which of the following represents $\Delta x$ in the general form of a Riemann sum?

Which definite integral corresponds to the limit: $\lim_{n \to \infty} \sum_{i=1}^{n} \frac{2}{n} \cdot (1 + \frac{2i}{n})^2$?

Consider the summation $\sum_{i=1}^{4} i^2$. What is the result of evaluating this summation?

A particle's velocity is given by $v(t) = t^2$ on the interval $[1, 3]$. Approximate the displacement using a Riemann sum with 2 subintervals and right endpoints.

What does the $\sum$ symbol represent in summation notation?

For the function $f(x) = x^2$ on the interval $[0, 4]$ with $n$ subintervals, what is $x_i$ using right endpoints?

Determine whether the summation $\sum_{i=0}^{n-1} \frac{1}{n}f(\frac{i}{n})$ represents a left or right Riemann sum.

Given $\lim_{n \to \infty} \sum_{i=1}^{n} \frac{2}{n} \cdot ((\frac{2i}{n})^3 + 1)$, identify the corresponding definite integral and evaluate it.

Describe how to set up a definite integral representing the area of the region bounded by the curve $f(x) = x^2$ and the x-axis from $x = 0$ to $x = 2$, and express it as a limit of Riemann sum.