Definite Integrals and Areas Under Curves
You're tasked with finding the area of a plot of land shaped like a curve. How would you do it?
This is where definite integrals come into play.
What is a Definite Integral?
A definite integral is a mathematical tool used to calculate the accumulated value of a function over a specific interval.
It is written as:
$$ \int_a^b f(x) \, dx $$
- $a$ and $b$ are the lower and upper limits of integration, respectively.
- $f(x)$ is the function being integrated.
- $dx$ indicates that the integration is with respect to $x$.
The definite integral $\int_a^b f(x) \, dx$ represents the net area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$.
The Fundamental Theorem of Calculus
- The Fundamental Theorem of Calculus connects differentiation and integration.
- It states that if $F(x)$ is an antiderivative of $f(x)$, then:
$$ \int_a^b f(x) \, dx = F(b) - F(a) $$
- $F(b) - F(a)$ represents the change in the antiderivative $F(x)$ between $x = a$ and $x = b$.
