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The integral $$\int\limits_{{{\text{x}}_1}}^{{{\text{x}}_2}} {{{\text{x}}^2}{\text{dx}}} $$  with x2 > x1 > 0 is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $$I$$ is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

The integral $$\int\limits_{{{\text{x}}_1}}^{{{\text{x}}_2}} {{{\text{x}}^2}{\text{dx}}} $$  with x2 > x1 > 0 is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $$I$$ is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship? Correct Answer J > $$I$$

Related Questions

With respect to the numerical evaluation of the definite integral $${\text{K}} = \int_{\text{a}}^{\text{b}} {{{\text{x}}^2}{\text{dx,}}} $$   where a and b are given, which of the following statements is/are TRUE?
I. The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II. The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.
Statement : It has been reported by one of the TV channels that the answer papers of Board examination of one State are evaluated by students studying in the same standard with the help of model answers as instructed by the examiners.

Courses of Action :
I. All such examiners should be immediately suspended from their official positions.
II. All such papers evaluated by the students should be immediately confiscated and got evaluated by qualified teachers.
III. The Board should explore possibilities even though they are remote, of getting the answer papers of this examination evaluated by computerised machines.
A common convention is: • is evaluated first • and are evaluated next • Quantifiers are evaluated next • is evaluated last.
A common convention is:
• is evaluated first
• and are evaluated next
• Quantifiers are evaluated next
• is evaluated last.
Consider the line integral $$I = \int_{\text{c}} {\left( {{{\text{x}}^2} + {\text{i}}{{\text{y}}^2}} \right){\text{dz,}}} $$    where z = x + iy. The line c is shown in the figure below
Complex Variable mcq question image
The value of $$I$$ is
In the following question, sentences of a paragraph have been jumbled and labeled as P, Q, R, and S. You are required to rearrange the jumbled sentences of the paragraph and mark your response accordingly by selecting the correct option. S1: There's nothing new in the idea that adolescence is a difficult period for everyone the adolescents themselves, of course, as well as for their parents P. Quite simply, the two generations live worlds apart. Now, psychologists are beginning to understand the biological basis for this eternal conflict. Q. Teenagers' brains really are different from those of their parents and a fully adult brain may not develop until well after the teenage years. R. For their parts, adolescents can't understand what their parents are worried about; they find their tastes and responsible concerns boring and outdated S. From an adult point of view, adolescents are difficult, rebellious, irresponsible, often selfish, and self-absorbed, prone to dangerous behaviour that risks their health, their life, and their possibility of future happiness. S6: This new knowledge, it is hoped, will help all of us deal better with the difficult and often painful years of transition from childhood to adulthood.
Read the passage carefully and select the best answer to each question out of the given four alternatives.
Public opinion may be passive and false or active and real. It is claimed in theory that all governments are ultimately based on the opinion or sanction of the governed. But we find that in practice the people's rights are often trodden down and tyranny and oppression are allowed to continue. The government does it not because the people want it to do so but because they are too idle, too uneducated and too disunited or timid to oppose the govern ment. Such public opinion is passive and false and not an active verdict. But when we find people alert, intelligent and determined to let the government know their will, when they want to exercise actively their voice in the management of their country, we have an instance of true or active public opinion. True public opinion is formed by and expressed through the press, the platform, political parties and educational institutions. These have sacred duties to perform, duties on which depends the ultimate good of the entire community. The press today wields a tremendous influence, So it should support the causes and move ments and condemn the wrong one's and thus teach people to form correct opinion. A free and fair press ventilates the grievances of the public. Thus a healthy relationship develops between the people and the government through out an unbiased press. Political parties also help to create and regulate opinions. No less important part is played by the educational Institutions which train the minds of the young people who will be the citizens of tomorrow. It has been said that modern Germany and China have been made by their universities. It is necessary that the young and the growing minds should imbibe the spirit of fellow-feeling, the spirit of tolerance, the habit of compromise, and show due regard for the feelings and opinion of others without which a democratic society cannot function, let alone succeed. When there is true awakening of the people, we shall have the real and conscious public opinion. And justice will reign on earth and truly will the voice of the people be the voice of God.
According to the passage, which of the following is needed from the people to have true or active public opinion? 1. Alertness
2. Determination
3. Intelligence
4. Raising their voice
Given below are six sentences. A and F are the first and the last sentences. The rest of the sentences namely B, C, D and E are jumbled. Select the option that gives their correct order. A. The well was highly useful to the people who lived in the surrounding huts. B. The well was dangerous: but it had been there for so long that no one worried about it. C. Consequently, the mouth of the well had gradually widened as the soil and rocks on the side fell in. D. There had never been any money for a wall around the well. E. They had dug it two years before because the only water tap in the colony was not enough for the slum’s growing population. F. Apart from a chicken that had fallen in and had been rescued, the well seemed harmless.
A 2nd degree polynomial, f(x) has values of 1, 4 and 15 at x = 0, 1 and 2, respectively. The integral $$\int\limits_0^2 {{\text{f}}\left( {\text{x}} \right){\text{dx}}} $$   is to be estimated by applying the trapezoidal rule to this data. What is the error (defined as "true value - approximate value") in the estimate?
Choose the set of figures which follows the given rule.
Rule: Closed figures losing their sides and open figures gaining their sides.
Rule Detection mcq question image