If x denotes the greatest integer function. Draw a rough sketch of the portions of the curves x 2 = 4 x y and y 2 = 4 y x that lie within the square x , y 1 ≤ x ≤ 4 , 1 ≤ y < 4 . Find the area of the part of the square that is enclosed by the two curves and the line x + y = 3 .

We know that x = 1 , 1 ≤ x < 4 2 , x = 4 & y = 1 , 1 ≤ y < 4 2 , y = 4 The equations of the given curves will be x 2 = 4 y ∀ x ∈ 1 , 4 & y 2 = 4 x ∀ y ∈ 1 , 4 . Let, A be the area bounded by the curves x 2 = 4 y , y 2 = 4 x & x + y = 3 . ⇒ A = ∫ 1 4 2 x d x - ∫ 2 4 x 2 4 d x - ∫ 1 2 3 - x d x ⇒ A = 2 x 3 2 3 2 1 4 - x 3 12 2 4 - 3 x - x 2 2 1 2 ⇒ A = 4 3 8 - 1 - 1 12 64 - 8 - 4 - 5 2 ⇒ A = 28 3 - 14 3 - 3 2 ⇒ A = 19 6

If x denotes the greatest integer function. Draw a rough sketch of the portions of the curves x 2 = 4 x y and y 2 = 4 y x that lie within the square x , y 1 ≤ x ≤ 4 , 1 ≤ y < 4 . Find the area of the part of the square that is enclosed by the two curves and the line x + y = 3 .

We know that x = 1 , 1 ≤ x < 4 2 , x = 4 & y = 1 , 1 ≤ y < 4 2 , y = 4 The equations of the given curves will be x 2 = 4 y ∀ x ∈ 1 , 4 & y 2 = 4 x ∀ y ∈ 1 , 4 . Let, A be the area bounded by the curves x 2 = 4 y , y 2 = 4 x & x + y = 3 . ⇒ A = ∫ 1 4 2 x d x - ∫ 2 4 x 2 4 d x - ∫ 1 2 3 - x d x ⇒ A = 2 x 3 2 3 2 1 4 - x 3 12 2 4 - 3 x - x 2 2 1 2 ⇒ A = 4 3 8 - 1 - 1 12 64 - 8 - 4 - 5 2 ⇒ A = 28 3 - 14 3 - 3 2 ⇒ A = 19 6

Let A B C be a triangle with vertices A 6 , 2 3 + 1 , B 4 , 2 and C 8 , 2 . If R be the region consisting of all these points and point P inside ∆ A B C which satisfy d P , B C ≥ max. d P , A B , d P , A C where d P , L denotes the distance of the point P from the line L . Sketch the region R and find its area.

B C = 8 - 4 2 + 0 2 = 4 unit A B = 6 - 4 2 + 2 3 + 2 - 2 2 = 4 unit A C = 8 - 6 2 + 2 - 2 3 - 2 2 = 4 unit ⇒ ∆ A B C is an equilateral triangle Consider, d P , B C = d P , A B ⇒ Point P should lie on angle bisector of angle B ∙ d P , B C - d P , A B ≥ 0 Point P should lie in shaded part ∙ d P , B C - d P , A C ≥ 0 Point P should lie in shaded part ∙ For point P to satisfy d P , B C ≥ max . d P , A B , d P , A C it should lie in shaded part Since ∆ A B C is equilateral, so angle bisector will also be altitudes & also median. ∙ So, required area = Area of quadrilateral A E Q D = 2 Area of ∆ A E Q = 2 1 2 × A E × E Q Since C E is median, so A E = 4 2 = 2 unit Q E = 2 tan 30 ° = 2 3 So, area of quadrilateral A E Q D = 2 1 2 × 2 × 2 3 = 4 3 = 4 3 3 sq. units

Area bounded by the line y = x , curve y = f x , f x > x ∀ x > 1 and the lines x = 1 , x = t is t + 1 + t 2 - 1 + 2 ∀ t > 1 . Find f x for x > 1 .

The area bounded by y = f x and y = x between the lines x = 1 and x = t is ∫ 1 t f x - x d x But it is equal to t + 1 + t 2 - 1 + 2 ∴ ∫ 1 t f x - x d x = t + 1 + t 2 - 1 + 2 Differentiating both sides w.r.t., t we get f t - t = 1 + t 1 + t 2 ⇒ f t = 1 + t + t 1 + t 2 or f x = 1 + x + x 1 + x 2 .

MEDIUM

JEE Main/Advance

IMPORTANT

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If $[x]$ denotes the greatest integer function. Draw a rough sketch of the portions of the curves $x^{2} = 4 [\sqrt{x}] y$ and $y^{2} = 4 [\sqrt{y}] x$ that lie within the square $\{(x, y) 1 \leq x \leq 4, 1 \leq y < 4\}$ . Find the area of the part of the square that is enclosed by the two curves and the line $x + y = 3$ .

Important Questions on Area under Curves

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-4>Q 4

Find the area of the region bounded by

y = f (x), y = |g (x)|

and the lines

x = 0, x = 2,

where

' f' and' g'

are continuous functions satisfying

f (x + y) = f (x) + f (y) - 8 x y \forall x, y \in R

and

g (x + y) = g (x) + g (y) + 3 x y (x + y) x, y \in R

also

f^{'} (0) = 8

and

g^{'} (0) = - 4

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-4>Q 5

Let

f (x) = \{\begin{cases} - 2 & , - 3 \leq x \leq 0 \\ x - 2 & , 0 < x \leq 3 \end{cases},

where

g (x) = \min \{f (| x |) + | f (x) |, f (| x |) - | f (x) |\}

Find the area bounded by the curve

g (x)

and the

x

-axis between the ordinates

x = 3

and

x = - 3 .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-4>Q 6

Find the area bounded by

y = [- 0.01 x^{4} - 0.02 x^{2}],

(where

[.]

G . l . F .

) and curve

3 x^{2} + 4 y^{2} = 12,

which lies below

y = - 1 .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-4>Q 7

Let

A B C

be a triangle with vertices

A (6, 2 (\sqrt{3} + 1)), B (4, 2)

and

C (8, 2) .

R

be the region consisting of all these points and point

P

inside

∆ A B C

which satisfy

d (P, B C) \geq

max.

\{d (P, A B), d (P, A C)\}

where

d (P, L)

denotes the distance of the point

P

from the line

L .

Sketch the region

R

and find its area.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-4>Q 8

Find the area of the region which contains all the points satisfying the condition

| x - 2 y | + | x + 2 y | \leq 8

and

x y \geq 2 .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-4>Q 9

Find the area of the region which is inside the parabola

y = - x^{2} + 6 x - 5,

out side the parabola

y = - x^{2} + 4 x - 3

and left of the straight line

y = 3 x - 15 .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-4>Q 10

Consider the curve $C : y = \sin 2 x - \sqrt{3} |\sin x|, C$ cuts the $x$ -axis at $(a, 0), x \in (- π, π)$

$A_{1} :$ The area bounded by the curve $C$ and the positive $x$ -axis between the origin and the line $x = a .$

$A_{2} :$ The area bounded by the curve $C$ and the negative $x$ -axis between the line $x = a$ and the origin. Prove that $A_{1} + A_{2} + 8 A_{1} A_{2} = 4 .$

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-4>Q 11

Area bounded by the line

y = x,

curve

y = f (x), (f (x) > x \forall x > 1)

and the lines

x = 1, x = t

t + \sqrt{1 + t^{2}} - (1 + \sqrt{2}) \forall t > 1 .

Find

f (x)

for

x > 1 .

If x denotes the greatest integer function. Draw a rough sketch of the portions of the curves x2=4xy and y2=4yx that lie within the square x,y 1≤x≤4, 1≤y<4. Find the area of the part of the square that is enclosed by the two curves and the line x+y=3.

Important Questions on Area under Curves

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