Antiderivatives, Riemann Sums & Definite Integrals

An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). It represents the reverse process of differentiation.

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then $$\int_{a}^{b} f(x)dx = F(b) - F(a)$$ .

The antiderivative is given by $$\int \frac{1}{x^n}dx = \frac{x^{-n+1}}{-n+1} + C$$ .

A Riemann Sum is a method for approximating the total area under a curve by dividing it into intervals and summing the areas of rectangles formed.

The types of Riemann Sums include Left Riemann Sum, Right Riemann Sum, and Midpoint Riemann Sum, depending on how the heights of the rectangles are determined.

The area under a curve can be approximated using the formula: $$A \approx \sum_{i=1}^{n} f(x_i^*) \Delta x$$ , where $$x_i^*$$ is a sample point in each interval.

The evaluation gives $$\frac{-5}{2x^2} + C$$ .

The constant C represents the family of all antiderivatives, as the derivative of a constant is zero.

A definite integral calculates the net area under a curve between two specified limits, denoted as $$\int_{a}^{b} f(x)dx$$ .

To approximate, calculate the height of the function at the left endpoint of each interval and sum the areas of the rectangles formed.