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The area enclosed between the parabolas $y^{2} = 16 x$ and $x^{2} = 16 y$ is

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Important Questions on Area under Curves

MEDIUM

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

Let

A = \{(x, y) : y^{2} \leq 4 x, y - 2 x \geq - 4\}

. The area of the region

A

in square units is

MEDIUM

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area of the region bounded by the lines

y = 2 x + 1, y = 3 x + 1

and

x = 4

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area enclosed by the curves

y = \sin x + \cos x

and

y = |\cos x - \sin x|

over the interval

[0, \frac{π}{2}]

MEDIUM

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area (in sq. units) of the region

A = \{(x, y) : x^{2} \leq y \leq x + 2\}

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area (in square units) bounded by the curves $y = \sqrt{x}, 2 y - x + 3 = 0$ , $X$ -axis and lying in the first quadrant is

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area (in

sq . units

) of the region

{(x, y) : y^{2} \geq 2 x

and

x^{2} + y^{2} \leq 4 x, x \geq 0, y \geq 0}

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area bounded by the curve

y = \cos x

, the line joining

(- \frac{π}{4}, \cos (- \frac{π}{4})) & (0, 2)

and the line joining

(\frac{π}{4}, \cos (\frac{π}{4})) & (0, 2)

MEDIUM

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

If the area (in sq. units) bounded by the parabola

y^{2} = 4 λ x

and the line

y = λ x, λ > 0,

\frac{1}{9}

, then

λ

is equal to

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The parabola

y^{2} = 4 x + 1

divides the disc

x^{2} + y^{2} \leq 1

into regions with areas

A_{1}

and

A_{2} .

Then

|A_{1} - A_{2}|

equals

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

On the real line

R

, we define two functions

f

and

g

as follows:

f (x) = m i n {x - [x], 1 - x + [x]}

g (x) = m a x \{x - [x], 1 - x + [x]\}

Where

[x]

denotes the largest integer not exceeding

x

. The positive integer

n

for which

\int_{0}^{n} (g (x) - f (x)) d x = 100

MEDIUM

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area (in sq. units) of the region

A = \{(x, y) : \frac{y^{2}}{2} \leq x \leq y + 4\}

is:

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

Area of the region

{(x, y) \in R^{2} : y \geq \sqrt{|x + 3|},

5 y \leq x + 9 \leq 15}

is equal to

MEDIUM

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area (in sq. units) of the region bounded by the curves

y = 2^{x}

and

y = |x + 1|

, in the first quadrant is

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area of the region

(

in square units

)

above the

x

-axis bounded by the curve

y = \tan x, 0 \leq x \leq \frac{π}{2}

and the tangent to the curve at

x = \frac{π}{4}

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

If the area (in sq. units) of the region

\{(x, y) : y^{2} \leq 4 x, x + y \leq 1, x \geq 0, y \geq 0\}

a \sqrt{2} + b,

then

a - b

is equal to

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

For

a > 0,

let the curves

C_{1} : y^{2} = a x

and

C_{2} : x^{2} = a y

intersect at origin

O

and a point

P .

Let the line

x = b (0 < b < a)

intersect the chord

O P

and the

x

-axis at points

Q

and

R,

respectively. If the line

x = b

bisects the area bounded by the curves,

C_{1}

and

C_{2},

and the area of

∆ O Q R = \frac{1}{2}

sq. units, then

a

satisfies the equation:

MEDIUM

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area of the region (in sq. units), enclosed by the circle

x^{2} + y^{2} = 2

which is not common to the region bounded by the parabola

y^{2} = x

and the straight line

y = x

, is

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area (in sq. units) of the region

\{(x, y) : x \geq 0, x + y \leq 3, x^{2} \leq 4 y a n d y \leq 1 + \sqrt{x}\}

MEDIUM

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

Let

I = \int_{a}^{b} (x^{4} - 2 x^{2}) d x .

I

is minimum then the ordered pair

(a, b)

HARD

Mathematics>Integral Calculus>Area under Curves>Area under the Curve

The area (in sq. units) of the region described by

[(x, y) : y^{2} \leq 2 x and y \geq 4 x - 1]

The area enclosed between the parabolas y2=16x and x2=16y is

Important Questions on Area under Curves

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The area enclosed between the parabolas $y^{2} = 16 x$ and $x^{2} = 16 y$ is