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TextBook Solutions for Punjab School Education Board Class 12 Mathematics Mathematics Part Ii Chapter 7 Integrals

Question 1

Evaluate $integral from 1 to 2 of straight x space dx$ as the limit of a sum.

Solution

Comparing $integral from 1 to 2 of straight x space dx space space with space integral from straight a to straight b of straight f left parenthesis straight x right parenthesis space dx comma space space we space get comma$
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$therefore space space space straight f left parenthesis straight a right parenthesis space equals space straight a comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight a plus straight h comma space space straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight a plus 2 straight h comma space...... straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight a plus left parenthesis straight n minus 1 right parenthesis straight h$
Now    $integral from straight a to straight b of straight f left parenthesis straight x right parenthesis space equals space stack Lt. straight h with straight h rightwards arrow 0 below left square bracket space straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis space plus space....... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
where n h = b - a
$rightwards double arrow$    $integral from straight a to straight b of xdx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight a plus straight a left parenthesis straight a plus straight h right parenthesis plus left parenthesis straight a plus 2 space straight h right parenthesis space plus space... plus left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket$
   $equals stack Lt space straight h with straight h rightwards arrow 0 below left square bracket left parenthesis straight a plus straight a plus straight a plus... to space straight n space terms right parenthesis space plus space straight h open curly brackets 1 plus 2 plus 3 plus.... plus left parenthesis straight n minus 1 right parenthesis close curly brackets right square bracket$
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$therefore space space space space space integral from straight a to straight b of xdx space equals space 1 half left parenthesis straight b squared minus straight a squared right parenthesis$
Put a = 1, b = 2
$therefore space space space space space space space space space space space space space integral from 1 to 2 of xdx space equals space 1 half left parenthesis 4 minus 1 right parenthesis space equals space 3 over 2$

Question 2

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Evaluate the following definite integrals as limit of a sum.
$integral from straight a to straight b of straight x space dx$

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Question 3

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Evaluate the following definite integral as limit of a sum.
$integral subscript 1 superscript 2 xdx$

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Question 4

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Evaluate the following definite integral as limit of a sum.
$integral subscript 0 superscript 5 left parenthesis straight x minus 1 right parenthesis dx$

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Question 5

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Evaluate the following integral as the limit of a sum:
$WiredFaculty$ $integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx$

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Question 6

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Evaluate
$integral subscript straight a superscript straight b straight x squared space dx$ as the limit of sum.

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Question 7

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Evaluate $integral subscript 1 superscript 3 straight x squared space dx$ as the limit of a sum.

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Question 8

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Evaluate $integral subscript 1 superscript 4 left parenthesis straight x squared minus straight x right parenthesis space dx$ as the limit of a sum.

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Question 9

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Evaluate $integral subscript 1 superscript 2 space left parenthesis 3 straight x squared plus 5 straight x right parenthesis space dx$ as limit of sums.

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Question 10

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Evaluate $integral subscript 1 superscript 3 space left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx$ as the limit of sums.

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TextBook Solutions for Punjab School Education Board Class 12 Mathematics Mathematics Part Ii Chapter 7 Integrals

Evaluate $integral from 1 to 2 of straight x space dx$ as the limit of a sum.

Evaluate the following definite integrals as limit of a sum.
$integral from straight a to straight b of straight x space dx$

Evaluate the following definite integral as limit of a sum.
$integral subscript 1 superscript 2 xdx$

Evaluate the following definite integral as limit of a sum.
$integral subscript 0 superscript 5 left parenthesis straight x minus 1 right parenthesis dx$

Evaluate the following integral as the limit of a sum:
$WiredFaculty$ $integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx$

Evaluate
$integral subscript straight a superscript straight b straight x squared space dx$ as the limit of sum.

Evaluate $integral subscript 1 superscript 3 straight x squared space dx$ as the limit of a sum.

Evaluate $integral subscript 1 superscript 4 left parenthesis straight x squared minus straight x right parenthesis space dx$ as the limit of a sum.

Evaluate $integral subscript 1 superscript 2 space left parenthesis 3 straight x squared plus 5 straight x right parenthesis space dx$ as limit of sums.

Evaluate $integral subscript 1 superscript 3 space left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx$ as the limit of sums.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

For Daily Free Study Material Join wiredfaculty

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Download Android App

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TextBook Solutions for Punjab School Education Board Class 12 Mathematics Mathematics Part Ii Chapter 7 Integrals

Evaluate as the limit of a sum.

Evaluate the following definite integrals as limit of a sum.

Evaluate the following definite integral as limit of a sum.

Evaluate the following definite integral as limit of a sum.

Evaluate the following integral as the limit of a sum:

Evaluate as the limit of sum.

Evaluate as the limit of a sum.

Evaluate as the limit of a sum.

Evaluate as limit of sums.

Evaluate as the limit of sums.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Evaluate $integral from 1 to 2 of straight x space dx$ as the limit of a sum.

Evaluate the following definite integrals as limit of a sum.
$integral from straight a to straight b of straight x space dx$

Evaluate the following definite integral as limit of a sum.
$integral subscript 1 superscript 2 xdx$

Evaluate the following definite integral as limit of a sum.
$integral subscript 0 superscript 5 left parenthesis straight x minus 1 right parenthesis dx$

Evaluate the following integral as the limit of a sum:
$WiredFaculty$ $integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx$

Evaluate
$integral subscript straight a superscript straight b straight x squared space dx$ as the limit of sum.

Evaluate $integral subscript 1 superscript 3 straight x squared space dx$ as the limit of a sum.

Evaluate $integral subscript 1 superscript 4 left parenthesis straight x squared minus straight x right parenthesis space dx$ as the limit of a sum.

Evaluate $integral subscript 1 superscript 2 space left parenthesis 3 straight x squared plus 5 straight x right parenthesis space dx$ as limit of sums.

Evaluate $integral subscript 1 superscript 3 space left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx$ as the limit of sums.