Given : and Then the area (in sq. units) of the region bounded b the curves, and between the lines, is :
Option: 1
Option: 2
Option: 3
Option: 4
Area Bounded by Curves When Intersects at More Than One Point 
Area bounded by the curves y = f(x), y = g(x) and intersect each other in the interval [a, b]
First find the point of intersection of these curves y = f(x) and y = g(x) , let the point of intersection be x = c
Area of the shaded region
When two curves intersects more than one point
rea bounded by the curves y=f(x), y=g(x) and intersect each other at three points at x = a, x = b amd x = c.
To find the point of intersection, solve f(x) = g(x).
For x ∈ (a, c), f(x) > g(x) and for x ∈ (c, b), g(x) > f(x).
Area bounded by curves,

Required area = Area of trapezium ABCD 
Correct Option (1)
View Full Answer(1)Let a function be continuous, and F be defined as : , where Then for the function F, the point is :
Option: 1 a point of infection.
Option: 2 a point of local maxima.
Option: 3 a point of local minima.
Option: 4 not a critical point.
Integration as Reverse Process of Differentiation 
Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given.
For example,
In the above example, the function cos x is the derived function of sin x. We say that sin x is an anti derivative (or an integral) of cos x. Similarly, x^{2} and e^{x} are the anti derivatives (or integrals) of 2x and e^{x} respectively.
Also note that the derivative of a constant (C) is zero. So we can write the above examples as:
Thus, anti derivatives (or integrals) of the above functions are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by selecting C arbitrarily from the set of real numbers.
For this reason C is referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function.
If the function F(x) is an antiderivative of f(x), then the expression F(x) + C is the indefinite integral of the function f(x) and is denoted by the symbol ∫ f(x) dx.
By definition,

Maxima and Minima of a Function 
Maxima and Minima of a Function
Let y = f(x) be a real function defined at x = a. Then the function f(x) is said to have a maximum value at x = a, if f(x) ≤ f(a) ∀ a ≥∈ R.
And also the function f(x) is said to have a minimum value at x = a, if f(x) ≥ f(a) ∀ a ∈ R
Concept of Local Maxima and Local Minima
The function f(x) is said to have a maximum (or we say that f(x) attains a maximum) at a point ‘a’ if the value of f(x) at ‘a’ is greater than its values for all x in a small neighborhood of ‘a’ .
In other words, f(x) has a maximum at x = ‘a’, if f(a + h) ≤ f(a) and f(a  h) ≤ f(a), where h ≥ 0 (very small quantity).
The function f(x) is said to have a minimum (or we say that f(x) attains a minimum) at a point ‘b’ if the value of f(x) at ‘b’ is less than its values for all x in a small neighborhood of ‘b’ .
In other words, f(x) has a maximum at x = ‘b’, if f(a + h) ≥ f(a) and f(a  h) ≥ f(a), where h ≥ 0 (very small quantity).

Correct Option (1)
View Full Answer(1)The value of is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
Properties of the Definite Integral (Part 2)  King's Property 
Property 4 (King's Property)
This is one of the most important properties of definite integration.

Application of Periodic Properties in Definite Integration 
Property 9
If f(x) is a periodic function with period T, then the area under f(x) for n periods would be n times the area under f(x) for one period, i.e.

View Full Answer(1)
is equal to
Option: 1
Option: 2
Option: 3
Option: 4
As we learnt
Walli's Method 
Definite integral by first principle
where
 wherein
View Full Answer(1)
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If where c is a constant of integration, then is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
Integration Using Substitution 
The method of substitution is one of the basic methods for calculating indefinite integrals.
Substitution  change of variable

Correct option (4)
View Full Answer(1)Let, and then is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
Integration as Reverse Process of Differentiation 
Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given.
For example,
In the above example, the function cos x is the derived function of sin x. We say that sin x is an anti derivative (or an integral) of cos x. Similarly, x^{2} and e^{x} are the anti derivatives (or integrals) of 2x and e^{x} respectively.
Also note that the derivative of a constant (C) is zero. So we can write the above examples as:
Thus, anti derivatives (or integrals) of the above functions are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by selecting C arbitrarily from the set of real numbers.
For this reason C is referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function.
If the function F(x) is an antiderivative of f(x), then the expression F(x) + C is the indefinite integral of the function f(x) and is denoted by the symbol ∫ f(x) dx.
By definition,

Trigonometric Identities 
Trigonometric Identities
These identities are the equations that hold true regardless of the angle being chosen.

Principal Value of function f1 (f (x)) 
Principal Value of function f^{1} (f (x))

Correct Option (4)
View Full Answer(1)For let the curves and intersect at origin O and a point P. Let the line intersect the chord OP and the xaxis at points Q and R,respectively. If the line bisects the are bounded by the curves, and and the area of , then 'a' satisfies the equation :
Option: 1
Option: 2
Option: 3
Option: 4
Parabola 
Parabola
A parabola is the set of all points (x, y) in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix in the plane.
Standard equation of a parabola
Let focus of parabola is S(a, 0) and directrix be x + a = 0, and axis as xaxis
P(x, y) is any point on the parabola.
Now, from the definition of the parabola,
which is the required equation of a standard parabola

Area Bounded by Curves When Intersects at More Than One Point 
Area bounded by the curves y = f(x), y = g(x) and intersect each other in the interval [a, b]
First find the point of intersection of these curves y = f(x) and y = g(x) , let the point of intersection be x = c
Area of the shaded region
When two curves intersects more than one point
rea bounded by the curves y=f(x), y=g(x) and intersect each other at three points at x = a, x = b amd x = c.
To find the point of intersection, solve f(x) = g(x).
For x ∈ (a, c), f(x) > g(x) and for x ∈ (c, b), g(x) > f(x).
Area bounded by curves,

By solving above you will get
Correct option (1)
View Full Answer(1)If , where C is a constant of integration, then is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
Indefinite integrals for Algebraic functions 
so
 wherein
Where
Integration by substitution 
The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.
 wherein
Since all variables must be converted into single variable , 
Let t =
t =
View Full Answer(1)
If f(a+b+1x)=f(x), for all x, where a and b are fixed positive real numbers, then is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
Properties of the Definite Integral (Part 2)  King's Property 
Property 4 (King's Property)
This is one of the most important properties of definite integration.

...........1
..........2
Adding 1 and 2
Correct Option (4)
View Full Answer(1)The area of the region, enclosed by the circle which is not common to the region bounded by the parabola and the straight line is :
Option: 1
Option: 2
Option: 3
Option: 4
Area Bounded by Curves When Intersects at More Than One Point 
Area bounded by the curves y = f(x), y = g(x) and intersect each other in the interval [a, b]
First find the point of intersection of these curves y = f(x) and y = g(x) , let the point of intersection be x = c
Area of the shaded region
When two curves intersects more than one point
rea bounded by the curves y=f(x), y=g(x) and intersect each other at three points at x = a, x = b amd x = c.
To find the point of intersection, solve f(x) = g(x).
For x ∈ (a, c), f(x) > g(x) and for x ∈ (c, b), g(x) > f(x).
Area bounded by curves,

Area between parabola and line is A1
Correct Option (2)
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