SL 5.11—Definite integrals, areas under curve onto x-axis and areas between curves Notes

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 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 $x = b$.

3. When the Curve Crosses the x-Axis

1. If $f(x)$ changes sign in $[a, b]$, the definite integral $\int_a^b f(x) \, dx$ gives the net area (positive area minus negative area).
2. To find the total area, split the integral at the points where $f(x) = 0$ and take the absolute value of each part.

Areas Between Curves

To find the area between two curves $y = f(x)$ and $y = g(x)$ over an interval $[a, b]$, use the formula:

$$ \int_a^b \left(f(x) - g(x)\right) \, dx $$

• $f(x)$ is the upper curve.
• $g(x)$ is the lower curve.

Analytical vs. Technological Approaches

While many integrals can be solved analytically, some require technology (e.g., graphing calculators or software) due to their complexity.