Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 1

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Area bounded by the region consisting of points $(x, y)$ satisfying $y \leq \sqrt{2 - x^{2}}, y^{2} \geq x, \sqrt{y} \geq - x$ is

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Important Questions on Area under Curves

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IMPORTANT

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

The area enclosed between the curves

y = \log_{e} (x + e), x = \log_{e} (\frac{1}{y})

and the

x

-axis is

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 3

The area bounded by the curve

x = {acos}^{3} t, y = a \sin^{3} t

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 5

The ratio in which the curve

y = x^{2}

divides the region bounded by the curve;

y = \sin (\frac{π x}{2})

and the

x

-axis as

x

varies from

0

to

1,

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 6

f (x) = \sin x, \forall x \in [0, \frac{π}{2}], f (x) + f (π - x) = 2 . \forall x \in (\frac{π}{2}, π]

and

f (x) = f (2 π - x), x \in (π, 2 π),

then the area enclosed by

y = f (x)

and

x

-axis is

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-2>Q 7

The area bounded by the curves

y = x e^{x}, y = x e^{- x}

and the line

x = 1

(in sq. units) is

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

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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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.