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Let I = Since <img class=Wirisformula style=vertical-align: -9px; src=/application/zrc/images/qvar/MAEN12061919-2.png alt=integral straight x squared dx space equals space straight x cubed over 3 space equals space straight F left parenthesis straight x right parenthesis comma space say. width=152 height=29 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«msup»«mi mathvariant=¨normal¨»x«/mi»«mn»2«/mn»«/msup»«mi»dx«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«msup»«mi mathvariant=¨normal¨»x«/mi»«mn»3«/mn»«/msup»«mn»3«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»F«/mi»«mo»(«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»)«/mo»«mo»,«/mo»«mo»§#160;«/mo»«mi»say«/mi»«mo».«/mo»«/math»> by second fundamental theorem, I = F(3) - F(2) = <img class=Wirisformula style=vertical-align: -9px; src=/application/zrc/images/qvar/MAEN12061919-4.png alt=fraction numerator left parenthesis 3 right parenthesis cubed over denominator 3 end fraction minus fraction numerator left parenthesis 2 right parenthesis cubed over denominator 3 end fraction space equals space 9 minus 8 over 3 space equals space 19 over 3 width=174 height=29 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»(«/mo»«mn»3«/mn»«msup»«mo»)«/mo»«mn»3«/mn»«/msup»«/mrow»«mn»3«/mn»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«mo»(«/mo»«mn»2«/mn»«msup»«mo»)«/mo»«mn»3«/mn»«/msup»«/mrow»«mn»3«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»9«/mn»«mo»-«/mo»«mfrac»«mn»8«/mn»«mn»3«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mn»19«/mn»«mn»3«/mn»«/mfrac»«/math»>

Evaluate the following definite integrals as limit of sums.
$integral subscript 2 superscript 3 straight x squared dx$

Let I = $integral subscript 2 superscript 3 straight x squared dx$
Since $integral straight x squared dx space equals space straight x cubed over 3 space equals space straight F left parenthesis straight x right parenthesis comma space say.$
$therefore$ by second fundamental theorem,
I = F(3) - F(2) = $fraction numerator left parenthesis 3 right parenthesis cubed over denominator 3 end fraction minus fraction numerator left parenthesis 2 right parenthesis cubed over denominator 3 end fraction space equals space 9 minus 8 over 3 space equals space 19 over 3$

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