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Integrals

Short Answer Type

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

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2. 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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3. Evaluate the following definite integral as limit of a sum.

integral subscript 1 superscript 2 xdx

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4. 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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5. Evaluate the following integral as the limit of a sum:

integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx

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6. Evaluate

integral subscript straight a superscript straight b straight x squared space dx

as the limit of sum.

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Long Answer Type

7.
Evaluate $integral subscript 1 superscript 3 straight x squared space dx$ as the limit of a sum.

Comparing $integral subscript 1 superscript 3 straight x squared dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space space we space get$
$straight f left parenthesis straight x right parenthesis space equals space straight x squared comma space space straight a space equals space 1 comma space space straight b space equals space 3$
$therefore space space space straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 1 space space space space space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 1 plus straight h right parenthesis space equals space left parenthesis 1 plus straight h right parenthesis squared space space space space straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals space left parenthesis 1 plus 2 straight h right parenthesis squared space space space straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space left parenthesis 1 plus stack straight n minus 1 with bar on top straight h right parenthesis squared$
Now, $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h close square brackets$
where n h = b - a = 3 - 1 = 2
$rightwards double arrow space space space integral subscript 1 superscript 3 straight x squared dx space equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 1 plus left parenthesis 1 plus straight h right parenthesis squared plus left parenthesis 1 plus 2 straight h right parenthesis squared plus.... plus left parenthesis 1 plus stack straight n minus 1 with bar on top straight h right parenthesis squared close square brackets$

$space equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket left parenthesis 1 plus 1 plus 1 plus...... space to space straight n space terms right parenthesis space plus space 2 space straight h space open curly brackets 1 plus 2 plus 3 plus... plus left parenthesis straight n minus 1 right parenthesis close curly brackets$
$plus space straight h squared space open curly brackets 1 squared plus 2 squared plus 3 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets right square bracket$

$equals space Lt with straight h rightwards arrow 0 below straight h open square brackets straight n plus 2 space straight h space fraction numerator left parenthesis straight n minus 1 right parenthesis straight n over denominator 2 end fraction plus straight h squared space fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below space open square brackets space straight n space straight h space plus space left parenthesis straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus space straight h right parenthesis plus fraction numerator left parenthesis space straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus space straight h right parenthesis over denominator 6 end fraction space close square brackets$

$equals space Lt with straight h rightwards arrow 0 below open square brackets 2 plus left parenthesis 2 right parenthesis left parenthesis 2 minus straight h right parenthesis plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis space left parenthesis 4 minus straight h right parenthesis over denominator 6 end fraction close square brackets space space space space space space open square brackets because space straight n space straight h space equals space 1 close square brackets equals space 2 plus left parenthesis 2 right parenthesis space left parenthesis 2 minus 0 right parenthesis space plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus 0 right parenthesis space left parenthesis 4 minus 0 right parenthesis over denominator 6 end fraction space equals 2 plus 4 plus 8 over 3 space equals space 26 over 3$

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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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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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10.

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.