Definite Integrals as Limit of a Sum A definite integral is a specific type of limit that involves finding the area under the curve of a function def...
A definite integral is a specific type of limit that involves finding the area under the curve of a function defined on a specific interval. Instead of calculating the exact area, we use limits to approximate it.
Definite integrals are evaluated using the limit of a sum. This means we break down the definite integral into a sum of infinitely many infinitesimal rectangles with widths and heights determined by the function.
Formally, the definite integral of a function f(x) with respect to x on the interval [a, b] is defined as:
**
where:
I represents the definite integral.
f(x) is the function to be integrated.
[a, b] is the interval of integration.
The limit of the sum of the areas of these rectangles as the number of rectangles approaches infinity is equal to the definite integral.
Example:
Let f(x) = x and [a, b] = [0, 4. Then:
This means the definite integral of f(x) with respect to x on the interval [0, 4] is equal to 2.
Applications of Definite Integrals:
Finding the area of a region under a curve.
Calculating the average value of a function.
Determining the volume of a 3D object by integrating the volume of its cross-sections.
Limitations of Definite Integrals:
Definite integrals only give an approximation of the area under the curve.
The accuracy of this approximation depends on the number of rectangles used in the sum.
Not all functions can be integrated using this method