If An is the area bounded by y=x and y=xn,n∈N, then A2.A3…An=

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If {A}_{{{n}}} is the area bounded by y=x and {y}={x}^{{{n}}},

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If $A_{n}$ is the area bounded by y=x and $y = x^{n}, n \in N,$ then $A_{2} . A_{3} \dots A_{n} =$

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Evelyn Discussed

A_{n}

is the area bounded by y=x and

y = x^{n}, n \in N,

then

A_{2} . A_{3} \dots A_{n} =

13 mins ago

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Practice questions from similar books

Question 1

The area enclosed by the curves

x y^{2} = a^{2} (a - x) and (a - x) y^{2} = a^{2} x

View solution

Question 2

a (a > 0)

is the value of parameter for each of which the are of the figure bounded by the straight line

y = \frac{a ^{2} - a x}{1 + a ^{4}}

and the parabola

y = \frac{x ^{2} + 2 a x + 3 a ^{2}}{1 + a ^{4}}

is the greatest, then the value of

a^{4}

is ___

View solution

Question 3

If the area bounded between x-axis and the graph of

y = 16 x - 3 x^{2}

between the ordinates

x = 1

and x = a is 19 square units, then take a can take the value

View solution

Question 4

If y=f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at

x = a, x = b, y = f (x) and y = f (c) (where c \in (a, b)) is minimum when c = \frac{a + b}{2}

Proof : A = a \int c (f (c) - f (x)) d x + c \int b (f (c)) d x

= f (c) (c - a) - a \int c (f (x)) d x + a \int b (f (x)) d x - f (c) (b - c)

\Rightarrow A = [2 c - (a + b)] f (c) + c \int b (f (x)) d x - a \int c (f (x)) d x

Differentiating w.r.t. c, we get

\frac{d A}{d c} = [2 c - (a + b)] f^{'} (c) + 2 f (c) + 0 - f (c) - (f (c) - 0)

For maxima and minima ,

\frac{d A}{d c} = 0

\Rightarrow f^{'} (c) [2 c - (a + b)] = 0 (a s f^{'} (c) \neq = 0)

Hence,

c = \frac{a + b}{2}

Also for c < \frac{a + b}{2}, \frac{d A}{d c} < 0 and for c > \frac{a + b}{2}, \frac{d A}{d c} > 0

Hence, A is minimum when

c = \frac{a + b}{2}

.
The value of the parameter a for which the area of the figure bounded by the abscissa axis, the graph of the function

y = x^{3} + 3 x^{2} + x + a

, and the straight lines, which are parallel to the axis of ordinates and cut the abscissa axis at the point of extremum of the function, which is the least, is

View solution

Question Text

A_{n}

is the area bounded by y=x and

y = x^{n}, n \in N,

then

A_{2} . A_{3} \dots A_{n} =

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If An​ is the area bounded by y=x and y=xn,n∈N, then A2​.A3​…An​=

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Practice questions from similar books

If $A_{n}$ is the area bounded by y=x and $y = x^{n}, n \in N,$ then $A_{2} . A_{3} \dots A_{n} =$