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Integrals

Long Answer Type

21.

Evaluate the following integrals as the limit of a sum
$integral subscript 0 superscript 3 left parenthesis straight x squared plus 2 straight x right parenthesis space dx$

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22.

Evaluate the following integrals as the limit of a sum
$integral subscript 0 superscript 3 left parenthesis 2 straight x squared plus 3 right parenthesis space dx$

102 Views

23.

Evaluate $integral subscript 2 superscript 3 space straight x cubed space dx$ as the limit of a sum.

153 Views

24. Evaluate

integral subscript straight a superscript straight b space straight e to the power of straight x space dx

as limit of sum.

97 Views

25.

Evaluate $integral subscript negative 1 end subscript superscript 1 space straight e to the power of straight x space dx$ as the limit of a sum.

231 Views

26.

Evaluate $integral subscript 0 superscript 2 straight e to the power of straight x space dx$ as the limit of a sum.

107 Views

27.

Evaluate $integral subscript 0 superscript 1 straight e to the power of 2 minus 3 straight x end exponent space dx$ as a limit of a sum.

101 Views

Short Answer Type

28.
Evaluate $integral subscript 0 superscript 3 straight e to the power of straight x space dx$ as the limit of sums.

Comparing $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space with space integral subscript 0 superscript 3 straight e to the power of straight x dx comma space we space get$
$straight f left parenthesis straight x right parenthesis space equals space straight e to the power of straight x comma space space space straight a space equals space 0 comma space space space straight b space equals space 3 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 0 right parenthesis space equals space straight e to the power of 0 space equals space 1 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis straight h right parenthesis space equals space straight e to the power of straight h straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis space 2 space straight h right parenthesis space equals space straight e to the power of 2 straight h end exponent$
.....................................................
   $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight f left parenthesis stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight e to the power of left parenthesis straight n minus 1 right parenthesis space straight h end exponent$
Now $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$rightwards double arrow$ $integral subscript 0 superscript 3 straight e to the power of straight x dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket 1 plus straight e to the power of straight h plus straight e to the power of 2 straight h end exponent plus.... plus straight e to the power of left parenthesis straight n minus 1 right parenthesis space straight h end exponent right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets fraction numerator 1 left parenthesis straight e to the power of straight n space straight h end exponent minus 1 right parenthesis over denominator straight e to the power of straight h minus 1 end fraction close square brackets$    $open square brackets because space space straight S subscript straight n space equals space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction close square brackets$

   $equals Lt with straight h rightwards arrow 0 below space straight h open square brackets fraction numerator straight e cubed minus 1 over denominator straight e to the power of straight h minus 1 end fraction close square brackets$    $open square brackets because space nh space equals space straight b minus straight a space equals space 3 minus 0 space equals space 3 close square brackets$

   $equals left parenthesis straight e cubed minus 1 right parenthesis space Lt with straight h rightwards arrow 0 below space fraction numerator 1 over denominator begin display style fraction numerator straight e to the power of straight h minus 1 over denominator straight h end fraction end style end fraction space equals space left parenthesis straight e cubed minus 1 right parenthesis. space 1 over 1 space equals space straight e cubed minus 1$

90 Views

29.

Evaluate $integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis space dx$ as the limit of a sum.