Properties of Definite Integrals Video Lecture | Mathematics for Competitive Exams

Ans. A definite integral is a mathematical concept that represents the signed area between the graph of a function and the x-axis over a specific interval. It is denoted by ∫\[a\]^\[b\] \[f(x)\] \[dx\], where \[f(x)\] is the integrand, \[a\] is the lower limit of integration, and \[b\] is the upper limit of integration.

Ans. The Fundamental Theorem of Calculus establishes a connection between the definite integral and the antiderivative of a function. It states that if \[F(x)\] is an antiderivative of \[f(x)\] on an interval \[(a, b)\], then \[\int_a^b f(x) dx = F(b) - F(a)\]. In other words, the definite integral of a function can be evaluated by finding the antiderivative of the function and subtracting the antiderivative values at the limits of integration.

Ans. The properties of definite integrals include linearity, additivity, and symmetry. - Linearity: The definite integral of a linear combination of functions is equal to the linear combination of their individual integrals. That is, \[\int_a^b (c_1f(x) + c_2g(x)) dx = c_1\int_a^b f(x) dx + c_2\int_a^b g(x) dx\]. - Additivity: The definite integral of the sum of two functions is equal to the sum of their individual integrals. That is, \[\int_a^b (f(x) + g(x)) dx = \int_a^b f(x) dx + \int_a^b g(x) dx\]. - Symmetry: If the limits of integration are symmetric about a point, the definite integral of an odd function over that interval is equal to zero. That is, \[\int_{-a}^a f(x) dx = 0\] for an odd function \[f(x)\].

Ans. Yes, the definite integral can be negative. The sign of the definite integral depends on the behavior of the function within the given interval. If the function lies below the x-axis or has negative values over the interval, the definite integral will be negative, representing the negative area between the function and the x-axis.

Ans. To find the area between two curves using the definite integral, one needs to determine the points of intersection between the curves. Let \[f(x)\] and \[g(x)\] be the two curves, and \[a\] and \[b\] be the x-coordinates of the points of intersection. The area between the curves can be calculated as \[\int_a^b |f(x) - g(x)| dx\]. By taking the absolute value of the difference between the two functions and integrating over the interval \[(a, b)\], the definite integral yields the desired area.