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Integrals

Question

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

Solution

Comparing $integral subscript straight a superscript straight b xdx space space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we space get$
$straight f left parenthesis straight x right parenthesis space equals space straight x$
$therefore$ $straight f left parenthesis straight a right parenthesis space equals space straight a plus straight h$
$straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight a plus 2 straight h$
......................................
$straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight a plus stack straight n minus 1 with bar on top straight h$
Now $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket$
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Integrals

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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