# RS Aggarwal Class 12 Definite integrals PDF

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### What are Definite Integrals?

A definite integral is a specific type of integral where the limits of integration are specified. It is used to calculate the signed area between a function and the x-axis over a specific interval. The definite integral of a function f(x) over the interval [a,b] is denoted by ∫b^a f(x)dx. The definite integral can be thought of as the limit of a summation of the function’s values, multiplied by the widths of small rectangles that cover the interval of integration.

The definite integral can be used to calculate the area under a curve, which is the area between the curve and the x-axis between two points a and b. This is often used in physics and engineering to calculate things like the distance traveled by an object or the amount of work done by a force.

The definite integral can also be used to calculate the average value of a function over an interval. This is often used in statistics to calculate the mean or median of a dataset.

The definite integral can be evaluated using the fundamental theorem of calculus, which states that if F(x) is the antiderivative of f(x) then the definite integral of f(x) over the interval [a,b] is equal to F(b) – F(a). This means that if we know the antiderivative of a function, we can use it to evaluate definite integrals.

The definite integral can also be evaluated using numerical methods such as the trapezoidal rule, Simpson’s rule, or the Gaussian quadrature. These methods involve approximating the area under the curve using a finite number of rectangles or other shapes and can be used when an antiderivative is not known.

It’s important to note that if you try to evaluate a definite integral where the upper limit is less than the lower limit, the integral will be negative, which means that the area is beneath the x-axis.

CH-16-Definite-Integrals

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