Here, in this chapter, we shall study a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabolas and. NCERT Notes for Class 12 Mathematics Chapter 8: Application of Integrals. Let f( x) be a function defined on the interval [a, b] and F(x) be its anti-derivative. Then. Mathematics Notes for Class 12 chapter 8. Application of Integrals. Let f(x) be a function defined on the interval [a, b] and F(x) be its anti-derivative. Then.

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Free PDF download of Class 12 Maths revision notes & short key-notes for Application of Integrals of Chapter 8 to score high marks in exams, prepared by. Applications of Integration. It is clear from (f(xi) − g(xi))∆x. This is exactly the sort of sum that turns into an integral in the limit, namely the integral. no effort, because in the limit these sums became definite integrals and definite integrals In many applications, the integral is viewed as net change over time.

Pre-calculus integration[ edit ] The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus ca. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was independently developed in China around the 3rd century AD by Liu Hui , who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere Shea ; Katz , pp. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. Further steps were made in the early 17th century by Barrow and Torricelli , who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers. Newton and Leibniz[ edit ] The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals.

The numbers and b are called limits of integration.

Properties of Definite Integral If at every point x of an interval [a, b] the inequalities g x? If m is the least value and M is the greatest value of the function f x on the interval [a, bl. If f is continuous on [a, b], then there exists a number c in [a, b] at which is called the mean value of the function f x on the interval [a, b].

If f22 x and g2 x are integrable on [a, b], then Let a function f x,? Then, for any?

If f t is an odd function, then is an even function. If f t is an even function, then is an odd function. If f t is an even function, then for non-zero a, is not necessarily an odd function. It will be an odd function, if If f x is continuous on [a,?

Integral Function Let f x be a continuous function defined on [a, b], then a function? Properties of Integral Function 1. The integral function of an integrable function is continuous.

Gamma Function If n is a positive rational number, then the improper integral is defined as a gamma function and it is denoted by? Area of Bounded Region The space occupied by the curve along with the axis, under the given condition is called area of bounded region. Volume of Solid Revolution 1. Curve Sketching 1.

Curve Sketching 1. If powers of y in a equation of curve are all even, then curve is symmetrical about Xaxis. If powers of x in a equation of curve are all even, then curve is symmetrical about Yaxis. When x is replaced by -x and y is replaced by -y, then curve is symmetrical in opposite quadrant.

Nature of Origin 1.

If point 0, 0 satisfies the equation, then curve passes through origin. If curve passes through origin, then equate low st degree term to zero and get equation of tangent.

If there are two tangents, then origin is a double point. Point of Intersection with Axes 1.

Now, find equation of tangent at this point i. Find regions where curve does not exists. Asymptotes 1. Equate coefficient of highest power of x and get asymptote parallel to X-axis. Similarly equate coefficient of highest power of y and get asymptote parallel to Y-axis.

It gives us the points where tngent is parallel or perpendicular to the X-axis. Points of Inflexion and solve the resulting equation. If some point of inflexion is there, then locate it exactly. We are not responsible for any type of mistake in data. All pdf files or link of pdf files are collected from various Resources Or sent by Students. If any pdf file have any copyright voilation please inform us we shell remove that file from our website.