Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 8

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Obtain the area enclosed by region bounded by the curves $y = x \ln x$ and $y = 2 x - 2 x^{2}$ .

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Important Questions on Area under Curves

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 10

Consider the following statements:

S_{1}

: The value of

\int_{0}^{2 π} \cos^{- 1} (\cos x) d x

is

π^{2}

S_{2}

: Area enclosed by the curve

|x - 2| + |y + 1| = 1

is equal to

3

sq. units

S_{3}

: If

\frac{d}{d x} f (x) = g (x)

for

a \leq x \leq b,

then

\int_{a}^{b} f (x) g (x) d x

equals to

\frac{{[f (b)]}^{2} - {[f (a)]}^{2}}{2} .

S_{4}

: Area of the region

R \equiv \{(x, y); x^{2} \leq y \leq x\}

is

\frac{1}{6}

State, in order, whether

S_{1}, S_{2}, S_{3}, S_{4}

are true or false

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 11

Find the value of

m (m > 0)

for which the area bounded by the line

y = m x + 2

and

x = 2 y - y^{2}

is

\frac{9}{2}

square units.

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 12

If the area bounded by $y = f^{- 1} (x), x = 10, x = 4$ and $x$ -axis given that area bounded by $y = f (x), x = 2, x = 6$ and $x$ -axis is $30 sq . units$ , where $f (2) = 4$ and $f (6) = 10$ is $k$ sq. units, then the value of $k$ is,

(given $f (x)$ is an invertible function)

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 13

Consider a line

ℓ : 2 x - \sqrt{3} y = 0

and a parametrised

C : x = \tan t, y = \frac{1}{\cos t} (0 \leq t < \frac{π}{2})

. If the area of part bounded by

ℓ, C

and the

y

-axis is equal to

\frac{1}{4} \ln (a + \sqrt{b})

, where

a, b, \in N, b

, is not perfect square then find the value of

(a + b)

.

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 14

Area bounded by

y = \sin^{- 1} x, y = \cos^{- 1} x, y = 0

in the first quadrant is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 15

Let

f (x)

be a non-negative, continuous and even function such that area bounded by

x

-axis,

y

-axis &

y = f (x)

is equal to

(x^{2} + x^{3})

square units

\forall x \geq 0,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 16

Let

' c'

be a positive real number such that area bounded by

y = 0, y = [\tan^{- 1} x]

from

x = 0

to

x = c

is equal to area bounded by

y = 0, y = [\cot^{- 1} x]

from

x = 0

to

x = c

(where

[*]

represents greatest integer function), then

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 17

Area bounded by

y = x^{2} - 2 |x|

and

y = - 1

is equal to