Integration and Accumulation of Change

Given the graph of $f(x)$, the derivative of $g(x)$, and the area between $f(x)$ and the x-axis from $x=a$ to $x=b$ is positive, what does this indicate about $g(x)$ over the interval $[a, b]$?

The graph of $f(x)$, the derivative of $g(x)$, consists of a quarter circle with radius 1 from $x = 0$ to $x = 1$ and a right triangle from $x = 1$ to $x = 3$. The triangle has a base of 2 and a height of -1. What is the value of $\int_{0}^{3} f(x) dx$?

A table gives values of a function $f(x)$ at $x = 0, 1, 2, 3, 4$. If we use a Left Riemann Sum with 4 equal subintervals to approximate $\int_{0}^{4} f(x) dx$, what values of $f(x)$ do we use to determine the heights of the rectangles?

Given an increasing function $f(x)$ on the interval $[a, b]$, which of the following Riemann Sums will always overestimate the area under the curve?

The velocity of a particle moving along the x-axis is given by $v(t)$, where $t$ is measured in seconds and $v(t)$ is measured in meters per second. The following data is collected:

Using a Riemann Sum with unequal subintervals, estimat...

Express the following definite integral as the limit of a Riemann Sum: $\int_{1}^{4} x^2 dx$

Identify the definite integral corresponding to the following limit of a Riemann Sum: $\lim_{n \to \infty} \sum_{i=1}^{n} (2 + \frac{3i}{n})^4 \frac{3}{n}$

Let $F(x) = \int_{0}^{x} (t^3 + 1) dt$. Find $F'(x)$.

Let $G(x) = \int_{1}^{x^2} \sqrt{t} dt$. Find $G'(x)$.

The graph of $f'(x)$ is given. On what interval(s) is $f(x)$ increasing?