Integrals

Calculus 1 - What Are Integrals? The Big Picture

An introduction to integral calculus explaining how integrals calculate total accumulation by finding the area under a curve using infinitely thin rectangles.

Calculus 1 - Riemann Sums: Approximating Area with Rectangles

Learn how Riemann sums approximate the area under a curve using rectangles. Covers left and right Riemann sums with examples and explains underestimation vs overestimation.

Calculus 1 - Riemann Sum from a Table

Learn how to approximate the area under a curve using a left Riemann sum with only a table of values, no graph needed.

Calculus 1 - Over- and Underestimation of Riemann Sums

Learn how left and right Riemann sums overestimate or underestimate area under curves depending on whether the function is increasing or decreasing.

Calculus 1 - Midpoint and Trapezoidal Sums

Learn midpoint and trapezoidal methods for approximating area under curves, which are more accurate than basic left or right rectangle sums.

Calculus 1 - Summation Notation (Sigma Notation)

Learn sigma notation, a mathematical shorthand for writing long sums. Understand the index, limits, and expressions to simplify repeated addition efficiently.

Calculus 1 - Riemann Sums in Sigma Notation

Learn how sigma notation simplifies Riemann sums by expressing the area of multiple rectangles in one compact formula, preparing for the definite integral concept.

Calculus 1 - The Definite Integral as a Limit of Riemann Sums

This video explains how definite integrals calculate exact area under curves by using infinitely many rectangles, building on the concept of Riemann sums.

Calculus 1 - Properties of Definite Integrals

Learn six essential properties of definite integrals that let you break apart, rearrange, and simplify integral calculations efficiently.

Calculus 1 - The Fundamental Theorem of Calculus

Explains the Fundamental Theorem of Calculus, showing how derivatives and integrals are connected and how to evaluate definite integrals using antiderivatives.

Calculus 1 - FTC with the Chain Rule

Learn how to apply the Fundamental Theorem of Calculus with the chain rule when the upper bound of an integral is a function of x, not just x itself.

Calculus 1 - Antiderivatives and Indefinite Integrals

Learn what antiderivatives are and how they reverse differentiation. Discover why the constant C is essential in indefinite integrals and how to write them using integral notation.

Calculus 1 - The Reverse Power Rule

Learn the reverse power rule for finding antiderivatives: add one to the exponent, divide by the new exponent, and add C. Includes examples with x⁴ and 6x².

Calculus 1 - Reverse Power Rule: Rewriting Before Integrating

Learn how to simplify complex integrals using algebra techniques like splitting fractions and converting roots to exponents before applying the power rule.

Calculus 1 - Antiderivatives of 1/x, eˣ, sin(x), and cos(x)

Learn antiderivative rules for 1/x, e^x, and trig functions by reversing derivative rules. Covers ln|x|, exponential, and sine/cosine integrals with visual confirmation.

Calculus 1 - Evaluating Definite Integrals with the FTC

Learn how to evaluate definite integrals using the Fundamental Theorem of Calculus: find the antiderivative, plug in bounds, and subtract.

Calculus 1 - Definite Integrals of Piecewise and Absolute Value Functions

Learn how to integrate piecewise and absolute value functions by splitting at formula changes. Example: evaluating the integral of |x| from -2 to 2 equals 4.

Calculus 1 - u-Substitution: The Concept

Learn u-substitution, a technique that reverses the chain rule to find antiderivatives by identifying inner functions and their derivatives within integrals.

Calculus 1 - u-Substitution: Indefinite Integrals Practice

Learn u-substitution for integrals step-by-step. Identify inner functions, match derivatives, substitute, integrate, and substitute back to solve complex integrals.

Calculus 1 - u-Substitution with Definite Integrals

Learn u-substitution for definite integrals. The key step is converting x-bounds to u-bounds to avoid errors and get the correct answer.

Calculus 1 - Integration Using Long Division

Learn how to integrate rational functions using polynomial long division when the numerator's degree is greater than or equal to the denominator's degree.

Calculus 1 - Accumulation Functions and Their Behavior

Learn how accumulation functions work using the Fundamental Theorem of Calculus. Discover how to sketch integral behavior from a graph without computing.

Calculus 1 - Integration Using Trig Identities

Learn how to integrate sine squared x using the double angle identity and when to use Pythagorean identity for odd powers of trig functions.