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CMI Entrance Exam

MT #5: Integral Calculus

Timings: 10:30-13:30 Hrs    Date: 21 March 2021

Part A: Short-answer type questions

Submission file: Write answers to all the ten questions on a single sheet of paper. Email a picture of your answer sheet. Name the file as PartA.jpg.

For this part, answers must be written without any explanation.

  1. Suppose is a fixed positive integer. If , what is the value of ?
  2. Evaluate .
  3. Let be a continuous real-valued function on for which Give an example of such a function.
  4. Consider the integral . Which of these statements are true?
    (a) The integral does not exist as the function is not defined at .
    (b) There is a removable singularity at but the integral does not exist.
    (c) There is a removable singularity at and the integral exists.
  5. Find the volume of the solid obtained when the region bounded by , and the line is revolved around the x-axis.

  6. Let be a function defined in the interval . Find the value of:
  7. For any real number , let denote the greatest integer such that . Find the value of .
  8. What is the area bounded by the curve , the x-axis and the straight line ?
  9. Suppose is a double differentiable function with and . What is the value of the integral below? If it possible for the integral to take multiple values, write Indeterminate.
  10. Compute the limit:

Part B: Subjective questions

Submission files: Each question in this part must be answered on a page of its own. Name the files as B1.jpg, B2.jpg, etc. In case you have multiple files for the same question, say B4, name the corresponding files as B4-1.jpg, B4-2.jpg, etc.

Clearly explain your entire reasoning. No credit will be given without reasoning. Partial solutions may get partial credit.

B1. Let be a continous function such that . Show that there exists such that .

B2. Let be a real-valued function defined on the interval . The following equations hold:

Prove that .

B3. Consider the parabola given by . The normal is constructed at a point and meets the parabola again in . Determine the coordinate of for which the arc length along the parabola between and is minimized.

B4. Let be a continous function defined on the interval with the property for in .
(a) Prove that: (b) Find a function for which equality holds in problem (a).

B5. Suppose . Prove that .

B6. Let . Prove that is bounded.