**12 class Maths Notes Chapter 8**

**Application of Integrals**

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**Application of Integrals**

**Notes class 12 maths**

**CBSE Revision Notes for CBSE Class 12 Mathematics Application of Integrals**Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).

Class 12 Maths Chapter-8

**Application of Integrals**

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**🔷 Chapter - 8 🔷**

**👉 Application of Integrals 👈**

**DEFINITE INTEGRATION**

**DEFINITION**

🔹 Let F (x) be any antiderivative of f(x), then for any two values of the independent variable x, say a and b, the difference F (b) – F (a) is called the definite integral of f (x) from a to b and is denoted by

The numbers a and b are called the limits of integration; a is the lower limit and b is the upper limit. Usually F (b) - F (a) is abbreviated by writing

**PROPERTIES OF DEFINITE INTEGRALS**

12. If f(x) is continuous on [a, b], m is the least and M is the greatest value of f (x) on [a, b], then

13. For any two functions f (x) and g (x), integrable on the interval [a, b], the Schwarz - Bunyakovsky inequality holds

14 . If a function f (x) is continuous on the interval [a, b], then there exists a point e ∈ (a, b) such that

**DIFFERENTIATION UNDER INTEGRAL SIGN**

**NEWTON LEIBNITZ'S THEOREM:**

🔹 If f is continuous on [a, b] and g(x) & h(x) are differentiable functions of x whose values lie in [a, b], then

**DEFINITE INTEGRAL AS A LIMIT OF SUM**

Let f (x) be a continuous real valued function defined on the closed interval [a, b] which is divided into n parts as shown in figure.

Let Sₙ denotes the area of these n rectangles.

Then, Sₙ = h f (a) + h f (a + h) + h f (a + 2h) +....... + h f (a + (n - 1) h)

Clearly, Sₙ is area very close to the area of the region bounded by curve y = f (x), x -axis and the ordinates x = a, x = b.

**REDUCTION FORMULAE IN DEFINITE INTEGRALS**

**AREA UNDER THE CURVES**

**AREA OF PLANE REGIONS**

1. The area bounded by the curve y =f (x), x-axis and the ordinates x = a. and x = b (where b> a) is given by

2 . The area bounded by the curve x =g (y), y- axis and the abscissae y = c and y = d (where d > c) is given by

3. If we have two curve y = f (x) and y = g (x), such that y = f (x) lies above the curve y = g (x) then the area bounded between them and the ordinates x= a and x = b (b > a), is given by

4 . The area bounded by the curves y = f (x) and y =g (x) between the ordinates x = a and x = b is given by

**5. CURVETRACING**

In order to find the area bounded by several curves, it is important to have rough sketch of the required portion. The following steps are very useful in tracing a cartesian curve f (x, y) = 0.

**Step 1: Symmetry**

(i) The curve is symmetrical about x-axis if all powers of y in the equation of the given curve are even.

(ii) The curve is symmetrical about y-axis if all powers of x in the equation of the given curve are even.

(iii) The curve is symmetrical about the line y = x, if the equation of the given curve remains unchanged on interchanging x and y.

(iv) The curve is symmetrical in opposite quadrants, if the equation of the given curve remains unchanged when x and y are replaced by x and y respectively.

**Step 2: Origin**

If there is no constant term in the equation of the algebraic curve, then the curve passes through the origin.

In that case, the tangents at the origin are given by equating to zero the lowest degree terms in the equation of the given algebraic curve.

For example, the curve y³ = x³ + axy passes through the origin and the tangents at the origin are given by axy = 0 i.e. x = 0 and y = 0.

**Step 3: Intersection with the Co-ordinate Axes**

(i) To find the points of intersection of the curve with X-axis, put y = 0 in the equation of the given curve and get the corresponding values of x.

(ii) To find the points of intersection of the curve with Y-axis, put x = 0 in the equation of the given curve and get the corresponding values of y.

**Step 4: Asymptotes**

Find out the asymptotes of the curve.

(i) The vertical asymptotes or the asymptotes parallel to y-axis of the given algebraic curve are obtained by equating to zero the coefficient of the highest power of y in the equation of the given curve.

(ii) The horizontal asymptotes or the asymptotes parallel to X-axis of the given algebraic curve are obtained by equating to zero the coefficient of the highest power of x in the equation of the given curve.

**Step 5: Region**

Find out the regions of the plane in which no part of the curve lies. To determine such regions we solve the given equation for y in terms of x or vice-versa. Suppose that y becomes imaginary for x > a, the curve does not lie in the region x > a.

**Step 6: Critical Points**

At such points y generally changes its character from an increasing function of x to a decreasing function of x or vice-versa.

Step 7: Trace the curve with the help of the above points.

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