$A = \int_{a}^{b} [f(x) - g(x)] dx$, where $f(x) \ge g(x)$ on $[a, b]$.

$A = \int_{0}^{2} (e^x - x) dx$

$A = \int_{a}^{b} |f(x) - g(x)| dx$

$\int e^x dx = e^x + C$

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$

$A = \int_{-2}^{B} [ln(x+3) - (x^4 + 2x^3)] dx$

$A = \int_{a}^{b} (top \ function - bottom \ function) dx$

$\int_{a}^{c} (f(x) - g(x)) dx + \int_{c}^{b} (g(x) - f(x)) dx$, where c is the intersection point.

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

$\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$

Subtracting the integrals effectively removes the area under the lower function from the area under the upper function, leaving only the area between them.

Intersection points define the limits of integration, indicating where the curves bound the area.

The 'top' function must be the minuend to ensure the area is positive, as area is always a positive quantity.

The result will be the negative of the actual area. Take the absolute value to correct.

The definite integral gives the signed area between a curve and the x-axis. For area between curves, it gives the area between the two curves.

Absolute value is needed when the top and bottom functions switch places within the interval of integration, ensuring the area is always positive.

The area between two curves is found by integrating the difference of the top and bottom functions over an interval [a, b].

It represents the area under a curve between two points on the x-axis.

You must divide the integral into multiple integrals, one for each region where the top and bottom functions remain the same.

It links differentiation and integration, allowing us to evaluate definite integrals using antiderivatives.

The area enclosed by two or more functions, found by integrating the difference between the functions over a given interval.

A definite integral calculates the area under a curve between two specified limits (bounds) on the x-axis.

The function with greater y-values over a given interval, used as the minuend in the integral.

The function with lesser y-values over a given interval, used as the subtrahend in the integral.

The x-values where the curves intersect, defining the interval over which the area is calculated.

Points where the graphs of two functions have the same x and y values.

A tool to find intersection points and visualize which function is 'on top'.

The area bounded by the intersection of two or more curves.

The function that is being integrated.

To find the numerical value of the definite integral, representing the area.