Let f : 0 , 1 → 0 , ∞ be a continuous function such that ∫ 0 1 f x d x = 10 . Which of the following statement is NOT necessarily true?

- 10 ≤ ∫ 0 1 sin ⁡ 100 x f x d x ≤ 10

MEDIUM

Earn 100

Among

$(S 1) : \lim_{n \to \infty} \frac{1}{n^{2}} (2 + 4 + 6 + \dots + 2 n) = 1$

$(S 2) : \lim_{n \to \infty} \frac{1}{n^{16}} (1^{15} + 2^{15} + 3^{15} + \dots + n^{15}) = \frac{1}{16}$

50% studentsanswered this correctly

Important Questions on Definite Integration

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

The value of

\lim_{n \to \infty} \prod_{r = 1}^{n} {(1 + \frac{r^{2}}{n^{2}})}^{\frac{2 r}{n^{2}}}

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

Let

f (x) = \lim_{n \to \infty} {(\frac{n^{n} (x + n) (x + \frac{n}{2}) \dots . . (x + \frac{n}{n})}{n! (x^{2} + n^{2}) (x^{2} + \frac{n^{2}}{4}) \dots . . (x^{2} + \frac{n^{2}}{n^{2}})})}^{\frac{x}{n}}

, for all

x > 0

. Then

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

Suppose the limit

L = \lim_{n \to \infty} \sqrt{n} \int_{0}^{1} \frac{1}{{(1 + x^{2})}^{n}} d x

exists and is larger than

\frac{1}{2}

, then

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

For each positive integer

n

, let

y_{n} = {\frac{1}{n} ((n + 1) (n + 2) ... (n + n))}^{\frac{1}{n}}

. If

\lim_{n \to \infty} y_{n} = L,

then the value of

[L]

(where

[x]

is the greatest integer less than or equal to

x

) is ____

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

Let

f : [0, 1] \to [0, \infty]

be a continuous function such that

\int_{0}^{1} f (x) d x = 10 .

Which of the following statement is NOT necessarily true?

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

a

and

b

are positive integers such that

b > a,

then

\lim_{n \to \infty} [\frac{1}{n a} + \frac{1}{n a + 1} + \frac{1}{n a + 2} + \dots + \frac{1}{n b}] =

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

\underset{n \to \infty}{l i m} (\frac{{(n + 1)}^{1 / 3}}{n^{4 / 3}} + \frac{{(n + 2)}^{1 / 3}}{n^{4 / 3}} + . . . . . + \frac{{(2 n)}^{1 / 3}}{n^{4 / 3}})

is equal to

EASY

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

Let

f

be a continuous function in

[0, 1]

, then

\lim_{n \to \infty} \sum_{j = 0}^{n} \frac{1}{n} f (\frac{j}{n})

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

\lim_{n \to \infty} {(\frac{(n + 1) (n + 2) \dots . 3 n}{n^{2 n}})}^{\frac{1}{n}}

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

The value of

I = \sum_{k = 1}^{98} \int_{k}^{k + 1} \frac{k + 1}{x (x + 1)} d x,

then

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

\lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + \dots + \sqrt{n}}{n^{\frac{3}{2}}} =

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

For $a \in R$ (the set of all real numbers), $a \neq - 1$ , $\lim_{n \to \infty} \frac{(1^{a} + 2^{a} + \dots + n^{a})}{{(n + 1)}^{a - 1} [(n a + 1) + (n a + 2) + \dots + (n a + n)]} = \frac{1}{60}$ . Then $a =$

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

\lim_{n \to \infty} \frac{3}{n} \{1 + \sqrt{\frac{n}{n + 3}} + \sqrt{\frac{n}{n + 6}} + \sqrt{\frac{n}{n + 9}} + \dots + \sqrt{\frac{n}{n + 3 (n - 1)}}\}

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

\int_{0}^{3} (2 + x^{2}) d x =

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

The value of

\lim_{n \to \infty} \frac{1}{n} \{\sec^{2} \frac{π}{4 n} + \sec^{2} \frac{2 π}{4 n} + \dots + \sec^{2} \frac{n π}{4 n}]

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

Let

I = \int_{0}^{1} \frac{x^{3} \cos 3 x}{2 + x^{2}} d x .

Then

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

Let

I = \int_{π / 4}^{π / 3} \frac{\sin x}{x} d x .

Then

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

For

a \in R, |a| > 1,

let

\lim_{n \to \infty} (\frac{1 + \sqrt[3]{2} + \dots + \sqrt[3]{n}}{n^{7 / 3} (\frac{1}{{(a n + 1)}^{2}} + \frac{1}{{(a n + 2)}^{2}} + \dots + \frac{1}{{(a n + n)}^{2}})}) = 54 .

Then the possible value(s) of

a

is/are:

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

\lim_{n \to \infty} (\frac{1^{a} + 2^{a} + \dots + n^{a}}{{(n + 1)}^{a - 1} [(n a + 1) + (n a + 2) + \dots + (n a + n)]}) = \frac{1}{60}

for some positive real number

a

, then

a

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Definite Integral as Limit of a Sum

\lim_{n \to \infty} (\frac{n}{n^{2} + 1^{2}} + \frac{n}{n^{2} + 2^{2}} + \frac{n}{n^{2} + 3^{2}} + . . \dots . + \frac{1}{5 n})

is equal to

Among S1:limn→∞1n2(2+4+6+…+2n)=1 S2:limn→∞1n16115+215+315+…+n15=116

Important Questions on Definite Integration

Learn and Prepare for any exam you want !

Among

$(S 1) : \lim_{n \to \infty} \frac{1}{n^{2}} (2 + 4 + 6 + \dots + 2 n) = 1$

$(S 2) : \lim_{n \to \infty} \frac{1}{n^{16}} (1^{15} + 2^{15} + 3^{15} + \dots + n^{15}) = \frac{1}{16}$