Definite integral

A definite integral is a mathematical concept that represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as $$\int_a^b f(x) \, dx$$ .

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then $$\int_a^b f(x) \, dx = F(b) - F(a)$$ .

To evaluate, find the antiderivative: $$\frac{1}{2}x^2 - 2x$$ . Then calculate: $$[\frac{1}{2}(3^2) - 2(3)] - [\frac{1}{2}(1^2) - 2(1)] = 0$$ .

To find the area, evaluate the definite integral $$\int_1^5 (3x^2 - 2x) \, dx$$ , which equals 100.

If a definite integral evaluates to zero, it indicates that the areas above and below the x-axis cancel each other out over the interval.

The integral evaluates to 0, as the function is odd and symmetric about the origin.

The geometric interpretation of a definite integral is the net area between the curve of the function and the x-axis over the specified interval.

A definite integral has specific limits and results in a numerical value, while an indefinite integral does not have limits and results in a family of functions plus a constant.

To find the area under a curve, set up the definite integral of the function over the desired interval and evaluate it.

The limits of integration define the interval over which the area is calculated and determine the bounds of the integral.