Trapezoidal rule for the evaluation of ∫ a b f x d x requires the interval a , b to be divided into:

Any number of subintervals of equal width

2 n subintervals of equal width

2 n + 1 subintervals of equal width

3 n subintervals of equal width

Trapezoidal rule for the evaluation of ∫ a b f x d x requires the interval a , b to be divided into:

Any number of subintervals of equal width

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

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Trapezoidal rule for the evaluation of $\int_{a}^{b} f (x) d x$ requires the interval $[a, b]$ to be divided into:

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Important Questions on Definite Integration

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Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

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>Estimation of Definite Integral

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>Estimation of Definite Integral

\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

HARD

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

\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

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

Let

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

Then

HARD

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

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>Estimation of Definite Integral

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>Estimation of Definite Integral

\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

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

Let

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

Then

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Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

Taking four subintervals, the value of

\int_{0}^{1} \frac{d x}{1 + x}

by using trapezoidal rule will be

HARD

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

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

is equal to

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

By Simpson's rule, the value of

\int_{1}^{2} \frac{d x}{x}

dividing the interval

(1, 2)

into four parts is

HARD

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

The value of

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

then

HARD

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

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 ____

HARD

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

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:

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

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

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

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

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

Dividing the interval

(1, 2)

into four equal parts and using Simpson's rule, the value of

\int_{1}^{2} \frac{d x}{x}

will be

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

Using trapezoidal rule and taking

n = 4

, the approximate value of integral

\int_{1}^{9} x^{2} d x

2 [\frac{1}{2} (1 + 9^{2}) + α^{2} + β^{2} + 7^{2}]

, then

EASY

Mathematics>Integral Calculus>Definite Integration>Estimation of Definite Integral

Simpson's rule for evaluation of

\int_{a}^{b} f (x) d x

requires the interval

(a, b)

to be divided into

Trapezoidal rule for the evaluation of ∫abfxdx requires the interval a,b to be divided into:

Important Questions on Definite Integration

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Trapezoidal rule for the evaluation of $\int_{a}^{b} f (x) d x$ requires the interval $[a, b]$ to be divided into: