Integral Calculus short conceptual (gemini)

What is an indefinite integral?

An integral without limits of integration, resulting in a function plus an arbitrary constant (+C).

What is a definite integral?

An integral with limits of integration (a and b), resulting in a single numerical value.

State the Fundamental Theorem of Calculus (Part 1).

The derivative of an antiderivative returns the original function: $\frac{d}{dx}{\int f(x)dx} = f(x)$.

State the Fundamental Theorem of Calculus (Part 2).

It relates the definite integral to the antiderivative: $\int_{a}^{b}f(x)dx = F(b) - F(a)$, where $F'(x) = f(x)$.

What is the geometric meaning of the definite integral?

The area under the curve of $f(x)$ from $x=a$ to $x=b$.

What is the relationship between $\int{a}^{b} f(x) dx$ and $\int{b}^{a} f(x) dx$?

$\int{a}^{b} f(x) dx = - \int{b}^{a} f(x) dx$ (Reversing limits changes the sign).

What is the value of $\int_{a}^{a} f(x) dx$?

Zero (The area from a point to itself is zero).

What is Integration?

The process of finding the antiderivative of a function.

What is the arbitrary constant of integration?

The constant C added to an indefinite integral, representing the family of functions that have the same derivative.

What technique is typically used to integrate products of functions like $x\sin(x)$?

Integration by Parts ($\int u dv = uv - \int v du$).

What technique is typically used for integrals involving $\sqrt{a^2 - x^2}$?

Trigonometric Substitution.

What is the method for integrating rational functions where the denominator is factorable?

Partial Fraction Decomposition.

What defines an Improper Integral?

An integral where one or both limits of integration are infinite, or the integrand has a discontinuity within the interval.