Definite Integral as a Limit of a Sum A definite integral , denoted by an Integral , represents a limit of a sum as the number of terms in the sum...
A definite integral, denoted by an Integral, represents a limit of a sum as the number of terms in the sum approaches infinity. In simpler terms, it tells us that the integral represents the area under a curve formed by the function being integrated.
Formally, the definite integral of a function f(x) with respect to x from a to b is defined as:
Integral from a to b of f(x) dx = lim_{n->∞} Σ_{i=1}^n f(a + i(b-a)/n}.
This means that the definite integral represents the limit as n approaches infinity of the sum of the function values evaluated at the endpoints of each subinterval of size (b-a) divided by n.
Example:
Let f(x) = x^2. Then, the definite integral from 0 to 1 of f(x) dx can be calculated as:
Integral from 0 to 1 of x^2 dx = lim_{n->∞} Σ_{i=1}^n (a + i(1-a)/n)^2.
Expanding the square, we get:
Integral from 0 to 1 of x^2 dx = lim_{n->∞} Σ_{i=1}^n (a^2 + 2ab - (a^2/n).
Simplifying the summation, we get:
Integral from 0 to 1 of x^2 dx = [a^2 + 2ab - (a^2/n)] from 1 to n.
As n approaches infinity, this approaches the area under the parabola y = x^2 with respect to x from 0 to 1, which is 1/3. Therefore, the definite integral of f(x) = x^2 with respect to x from 0 to 1 is 1/3.
Key points to understand:
The definite integral represents the limit of a sum.
It tells us the area under a curve formed by the function.
The integral converges as n approaches infinity.
The definite integral is used to evaluate definite integrals.
Further exploration:
Explore the properties of definite integrals, such as linearity and the fundamental theorem of calculus.
Practice evaluating definite integrals with different functions and limits.
Apply definite integrals to real-world problems involving area, displacement, and other physical concepts