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Consider two curves $C_{1} : y = \frac{1}{x} and C_{2} : y =$ $\ln x$ on the $x y$ plane. Let D₁ denotes the region surrounded by C₁, C₂ and the line $x = 1$ and D₂ denotes the region surrounded by C₁, C₂and the line $x = a$ . If D₁ = D₂. Find the value of ' $a$ '

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Important Questions on Area under Curves

HARD

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

Consider a region $R = \{(x, y) \in R^{2} : x^{2} \leq y \leq 2 x\}$ . If a line $y = α$ divides the area of region $R$ into two equal parts, then which of the following is true?

EASY

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

Area lying between the parabola

y^{2} = 4 a x

and its latus rectum is

HARD

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

Given:

f (x) = \{\begin{matrix} \begin{matrix} x, 0 \leq x < \frac{1}{2} \\ \frac{1}{2}, x = \frac{1}{2} \end{matrix} \\ 1 - x, \frac{1}{2} < x \leq 1 \end{matrix}

and

g (x) = {(x - \frac{1}{2})}^{2}, x \in R .

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

y = f (x)

and

y = g (x)

between the lines

2 x = 1

and

2 x = \sqrt{3},

is:

HARD

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

Let the functions

f : R \to R

and

g : R \to R

be defined by

f (x) = e^{x - 1} - e^{- |x - 1|}

and

g (x) = \frac{1}{2} (e^{x - 1} + e^{1 - x})

.
Then the area (in sq. units) of the region in the first quadrant bounded by the curves

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

and

x = 0

HARD

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

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

A = \{(x, y) : x^{2} + y^{2} \leq 1 and y^{2} \leq 1 - x\}

HARD

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

The area (in sq. units) of the region

A = \{(x, y) : | x | + | y | \leq 1, 2 y^{2} \geq | x |\}

MEDIUM

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

The area (in sq. units) of the region

A = \{(x, y) \in R \times R| 0 \leq x \leq 3, 0 \leq y \leq 4, y \leq x^{2} + 3 x\}

HARD

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

Calculate the area enclosed by the curves

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

MEDIUM

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

If the area of the region bounded by the curves,

y = x^{2}, y = \frac{1}{x}

and the lines

y = 0

and

x = t (t > 1)

1 sq . unit

, then

t

is equal to

HARD

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

The area (in sq. units) of the smaller portion enclosed between the curves,

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

and

y^{2} = 3 x

, is:

MEDIUM

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

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

a x \pm b y \pm c = 0 (a, b, c \in R

are positive and distinct) is

EASY

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

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

y = x^{2} + 2

and the lines,

y = x + 1, x = 0

and

x = 3

, is

MEDIUM

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

The area (in sq. units) of the region

\{(x - y) : \frac{1}{2} \leq y \leq \sin x, 0 \leq x \leq π\}

MEDIUM

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

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

y = x^{2} - 1

and

y = 1 - x^{2}

is equal to:

HARD

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

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

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

MEDIUM

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

Area of one loop formed by

| y | = | \sin x |

is (in sq. units)

MEDIUM

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

The larger of the two areas (in sq. units) into which the circle $x^{2} + y^{2} = 16 a^{2}$ is divided by the parabola $y^{2} = 6 a x$ , is

HARD

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

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

A = \{(x, y) | y \geq x^{2} - 5 x + 4, x + y \geq 1, y \leq 0\}

MEDIUM

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

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

y + 2 x^{2} = 0

and

y + 3 x^{2} = 1

, is equal to

HARD

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

If the area enclosed between the curves

y = k x^{2}

and

x = k y^{2}, (k > 0),

1 s q . u n i t .

Then

k

Consider two curves C 1 : y = 1 x and C 2 : y = lnx on the xy plane. Let D1 denotes the region surrounded by C1, C2 and the line x=1 and D2 denotes the region surrounded by C1, C2 and the line x=a. If D1 = D2. Find the value of 'a'

Important Questions on Area under Curves

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Consider two curves $C_{1} : y = \frac{1}{x} and C_{2} : y =$ $\ln x$ on the $x y$ plane. Let D₁ denotes the region surrounded by C₁, C₂ and the line $x = 1$ and D₂ denotes the region surrounded by C₁, C₂and the line $x = a$ . If D₁ = D₂. Find the value of ' $a$ '