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=Chapter 5: Continuity and Differentiability

=Chapter 1: Relations and Functions

=Chapter 2: Inverse Trigonometric Functions

=Chapter 3: Matrices

=Chapter 4: Determinants

=Chapter 6: Application of Derivatives

=Chapter 7: Integrals

=Chapter 8: Application of Integrals

=Chapter 9. Differential Equations

=Chapter 10: Vector Algebra

=Chapter 12: Linear Programming

= Chapter 11: Three Dimensional Geometry

 

Chapter 8: Application of Integrals

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,

Application of Integrals

The above is called the second fundamental theorem of calculus.

Application of Integralsis defined as the definite integral of f(x) from x = a to x = b. The numbers and b are called limits of integration. We write

Application of Integrals

Evaluation of Definite Integrals by Substitution

Consider a definite integral of the following form

Evaluation of Definite Integrals by Substitution

Step 1 Substitute g(x) = t

g ‘(x) dx = dt

Step 2 Find the limits of integration in new system of variable i.e.. the lower limit is g(a) and the upper limit is g(b) and the g(b) integral is nowEvaluation of Definite Integrals by Substitution

Step 3 Evaluate the integral, so obtained by usual method.

Properties of Definite Integral

properties of Definite Integral

13. Leibnitz Rule for Differentiation Under Integral Sign

(a) If Φ(x) and ψ(x) are defined on [a, b] and differentiable for every x and f(t) is continuous, then

Leibnitz Rule for Differentiation Under Integral Sign

(b) If Φ(x) and ψ(x) are defined on [a, b] and differentiable for every x and f(t) is continuous, then

Leibnitz Rule for Differentiation Under Integral Sign

14. If f(x) ≥ 0 on the interval [a, b], thenLeibnitz Rule for Differentiation Under Integral Sign

15. If (x) ≤ Φ(x) for x ∈ [a, b], then Leibnitz Rule for Differentiation Under Integral Sign

16. If at every point x of an interval [a, b] the inequalities g(x) ≤ f(x) ≤ h(x) are fulfilled, then

Leibnitz Rule for Differentiation Under Integral Sign

18. If m is the least value and M is the greatest value of the function f(x) on the interval [a, bl. (estimation of an integral), then

Leibnitz Rule for Differentiation Under Integral Sign

19. If f is continuous on [a, b], then there exists a number c in [a, b] at which

Leibnitz Rule for Differentiation Under Integral Sign

is called the mean value of the function f(x) on the interval [a, b].

20. If f22 (x) and g2 (x) are integrable on [a, b], then

Leibnitz Rule for Differentiation Under Integral Sign

21. Let a function f(x, α) be continuous for a ≤ x ≤ b and c ≤ α ≤ d.
Then, for any α ∈ [c, d], if

Leibnitz Rule for Differentiation Under Integral Sign

22. If f(t) is an odd function, thenLeibnitz Rule for Differentiation Under Integral Signis an even function.

23. If f(t) is an even function, thenLeibnitz Rule for Differentiation Under Integral Signis an odd function.

24. If f(t) is an even function, then for non-zero a,Leibnitz Rule for Differentiation Under Integral Signis not necessarily an odd function. It will be an odd function, if

Leibnitz Rule for Differentiation Under Integral Sign

25. If f(x) is continuous on [a, α], thenLeibnitz Rule for Differentiation Under Integral Signis called an improper integral and is defined asLeibnitz Rule for Differentiation Under Integral Sign

Leibnitz Rule for Differentiation Under Integral Sign

27. Geometrically, for f(x) > 0, the improper integralLeibnitz Rule for Differentiation Under Integral Signgives area of the figure bounded by the curve y = f(x), the axis and the straight line x = a.

Integral Function

Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined byIntegral Functionis called the integral function of the function f.

Properties of Integral Function

1. The integral function of an integrable function is continuous.
2. If φ(x) is the integral function of continuous function, then φ(x) is derivable and of φ ‘ = f(x) for all x ∈ [a, b].

Gamma Function

If n is a positive rational number, then the improper integralGamma Functionis defined as a gamma function and it is denoted by Γn
Gamma Function

Properties of Gamma Function

Properties of Gamma Function

walli's formula

Summation of Series by Definite Integral

Summation of Series by Definite Integral

The method to evaluate the integral, as limit of the sum of an infinite series is known as integration by first principle.

Important results

Area of Bounded Region

The space occupied by the curve along with the axis, under the given condition is called area of bounded region.

(i) The area bounded by the curve y = F(x) above the X-axis and between the lines x = a, x = b is given by Area of Bounded Region

Area of Bounded Region

(ii) If the curve between the lines x = a, x = b lies below the X-axis, then the required area is given by

Area of Bounded Region

(iii) The area bounded by the curve x = F(y) right to the Y-axis and the lines y = c, y = d is given by

Area of Bounded Region

(iv) If the curve between the lines y = c, y = d left to the Y-axis, then the area is given by

Area of Bounded Region

(v) Area bounded by two curves y = F (x) and y = G (x) between x = a and x = b is given by

Area of Bounded Region

(vi) Area bounded by two curves x = f(y) and x = g(y) between y=c and y=d is given byArea of Bounded Region

(vii) If F (x) ≥. G (x) in [a, c] and F (x) ≤ G (x) in [c,d], where a < c < b, then area of the region bounded by the curves is given as

Area of Bounded Region

Area of Curves Given by Polar Equations

Let f(θ) be a continuous function, θ ∈ (a, α), then the are t bounded by the curve r = f(θ) and <β) is

Area of Curves Given by Polar Equations

Area of Parametric Curves

Let x = φ(t) and y = ψ(t) be two parametric curves, then area bounded by the curve, X-axis and ordinates x = φ(t1), x = ψ(t2) is

Area of Parametric Curves

Volume and Surface Area

If We revolve any plane curve along any line, then solid so generated is called solid of revolution.

1. Volume of Solid Revolution

1. The volume of the solid generated by revolution of the area bounded by the curve y = f(x), the axis of x and the ordinatesVolume of Solid Revolutionit being given that f(x) is a continuous a function in the interval (a, b).
2. The volume of the solid generated by revolution of the area bounded by the curve x = g(y), the axis of y and two abscissas y = c and y = d isVolume of Solid Revolutionit being given that g(y) is a continuous function in the interval (c, d).

Surface of Solid Revolution

(i) The surface of the solid generated by revolution of the area bounded by the curve y = f(x), the axis of x and the ordinatesSurface of Solid Revolutionis a continuous function in the interval (a, b).

(ii) The surface of the solid generated by revolution of the area bounded by the curve x = f (y), the axis of y and y = c, y = d isSurface of Solid Revolutioncontinuous function in the interval (c, d).

Curve Sketching
1. symmetry

1. If powers of y in a equation of curve are all even, then curve is symmetrical about Xaxis.
2. If powers of x in a equation of curve are all even, then curve is symmetrical about Yaxis.
3. When x is replaced by -x and y is replaced by -y, then curve is symmetrical in opposite quadrant.
4. If x and y are interchanged and equation of curve remains unchanged curve is symmetrical about line y = x.

2. Nature of Origin

1. If point (0, 0) satisfies the equation, then curve passes through origin.
2. 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.

3. Point of Intersection with Axes

1. Put y = 0 and get intersection with X-axis, put x = 0 and get intersection with Y-axis.
2. Now, find equation of tangent at this point i. e. , shift origin to the point of intersection and equate the lowest degree term to zero.
3. Find regions where curve does not exists. i. e., curve will not exit for those values of variable when makes the other imaginary or not defined.

4. Asymptotes

1. Equate coefficient of highest power of x and get asymptote parallel to X-axis.
2. Similarly equate coefficient of highest power of y and get asymptote parallel to Y-axis.

5. The Sign of (dy/dx)

Find points at which (dy/dx) vanishes or becomes infinite. It gives us the points where tngent is parallel or perpendicular to the X-axis.

6. Points of Inflexion

Points of Inflexion.and solve the resulting equation.If some point of inflexion is there, then locate it exactly.

Taking in consideration of all above information, we draw an approximate shape of the curve

Shape of Some Curves is Given Below

shape of some curves

shape of some curves

shape of some curves


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Math nots For Class 12 Download PDF Application of Integrals Chapter 8:
Math nots For Class 12 Download PDF Application of Integrals Chapter 8:

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