Integration and Accumulation of Change

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

Given $\int_{0}^{3} f(x) dx = 5$ and $\int_{0}^{7} f(x) dx = 12$, find $\int_{7}^{3} f(x) dx$.

Given that $f(2) = 5$ and $\int_{2}^{6} f'(x) dx = 12$, find $f(6)$.

Find the indefinite integral: $\int (x^3 + 2x - 5) dx$

Evaluate the definite integral $\int_{0}^{\frac{\pi}{2}} sin(x) cos^2(x) dx$ using u-substitution.

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: $\int_{1}^{4} x^2 dx$

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?

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