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

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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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Important Questions on Area under Curves

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-3>Q 1

Area of the region bounded by the curve

y = e^{x}

and lines

x = 0

and

y = e

is

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

Let the straight line

x = b

divide the area enclosed by

y = {(1 - x)}^{2}, y = 0

and

x = 0

into two parts

R_{1} (0 \leq x \leq b)

and

R_{2} (b \leq x \leq 1)

such that

R_{1} - R_{2} = \frac{1}{4}

. Then

b

equals

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-3>Q 5

The area enclosed by the curves

y = \sin x + \cos x

and

y = | \cos x - \sin x |

over the interval

[0, \frac{π}{2}]

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-3>Q 7

If the line

x = α

divides the area of region

R = \{(x, y) \in R^{2} : x^{3} \leq y \leq x, 0 \leq x \leq 1\}

into two equal parts, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-3>Q 8

A farmer

F_{1}

has a land in the shape of a triangle with vertices at

P (0, 0), Q (1, 1)

and

R (2, 0)

. From this land, a neighbouring farmer

F_{2}

takes away the region which lies between the side

P Q

and a curve of the form

y = x^{n} (n > 1)

. If the area of the region taken away by the farmer

F_{2}

is exactly

30 %

of the area of

∆ P Q R

, then the value of

n

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-3>Q 9

The area of the region bounded by the parabola

{(y - 2)}^{2} = x - 1

, the tangent to the parabola at the point

(2, 3)

and the

x -

aixs is

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-3>Q 10

The area bounded by the curves

y = \cos x

and

y = \sin x

between the ordinates

x = 0

and

x = \frac{3 π}{2}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 31 - Area under Curves>Exercise-3>Q 11

The area of the region enclosed by the curves

y = x, x = e, y = \frac{1}{x}

and the positive

x

-axis is