Filter By

## All Questions

#### 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

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)

#### 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, x2 and ex  are the anti derivatives (or integrals) of 2x and ex 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)

## Crack CUET with india's "Best Teachers"

• HD Video Lectures
• Unlimited Mock Tests
• Faculty Support

#### 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.

-

#### is equal to Option: 1 $\dpi{150} \frac{18}{e^{4}}$ Option: 2 $\dpi{150} \frac{27}{e^{2}}$ Option: 3 $\dpi{150} 3\log ^{3-2}$ Option: 4 $\dpi{150} 3\log ^{3-2}$

As we learnt

Walli's Method -

Definite integral by first principle

where

- wherein

## Crack NEET with "AI Coach"

• HD Video Lectures
• Unlimited Mock Tests
• Faculty Support

#### 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)

#### 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, x2 and ex  are the anti derivatives (or integrals) of 2x and ex 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 f-1 (f (x)) -

Principal Value of function f-1 (f (x))

-

Correct Option (4)

## Crack JEE Main with "AI Coach"

• HD Video Lectures
• Unlimited Mock Tests
• Faculty Support

#### For let the curves and intersect at origin O and a point P. Let the line intersect the chord OP and the x-axis 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 x-axis

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

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)

#### 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 =

## Crack CUET with india's "Best Teachers"

• HD Video Lectures
• Unlimited Mock Tests
• Faculty Support

#### If  f(a+b+1-x)=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

Correct Option (4)

#### 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

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)