Theorem: the definite integral of a continuous function on [a,b]
The theorem we’re discussing is known as the Fundamental Theorem of Calculus
The theorem we’re discussing is known as the Fundamental Theorem of Calculus. It states that if a function f(x) is continuous on the interval [a, b], then the definite integral of f(x) with respect to x over [a, b] is equal to the antiderivative of f(x) evaluated at the upper limit of integration minus the antiderivative of f(x) evaluated at the lower limit of integration.
To put it in mathematical terms, if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b can be expressed as:
∫[a to b] f(x) dx = F(b) – F(a)
Where ∫ represents the integral, f(x) is the continuous function being integrated, dx indicates the variable of integration, and F(x) is an antiderivative of f(x).
Let’s go through an example to illustrate how to apply the Fundamental Theorem of Calculus:
Example: Find the definite integral of the function f(x) = 2x on the interval [1, 4].
Solution:
First, we need to find an antiderivative of f(x). In this case, the antiderivative of 2x with respect to x is x^2.
Next, we evaluate the antiderivative at the upper and lower limits of integration:
F(b) = (b^2) = (4^2) = 16
F(a) = (a^2) = (1^2) = 1
Finally, we subtract the result of F(a) from F(b) to find the definite integral:
∫[1 to 4] 2x dx = F(4) – F(1) = 16 – 1 = 15
Therefore, the definite integral of the function f(x) = 2x on the interval [1, 4] is 15.
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