Question

Express the limits as definite integral. lim_(norm(p rarr 0))sum_(k=1)^n(1/c_k)Deltax_k, where P is a partition of[1,4]

Applications of integrals
ANSWERED
asked 2020-10-20

Express the limits as definite integral.
\(\displaystyle\lim_{{{\left\|{{p}\rightarrow{0}}\right\|}}}{\sum_{{{k}={1}}}^{{n}}}{\left(\frac{{1}}{{c}_{{k}}}\right)}\Delta{x}_{{k}},{w}{h}{e}{r}{e}\ {P}\ {i}{s}\ {a}\ {p}{a}{r}{t}{i}{t}{i}{o}{n}\ {o}{f}{\left[{1},{4}\right]}\)

Answers (1)

2020-10-21
Step 1
Given
\(\displaystyle\lim_{{{\left\|{{p}\rightarrow{0}}\right\|}}}{\sum_{{{k}={1}}}^{{n}}}{\left(\frac{{1}}{{c}_{{k}}}\right)}\Delta{x}_{{k}}\)
Step 2
To express limits as a definite integrals.
The definition of definite integral is,
\(\displaystyle{\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}=\lim_{{{n}\rightarrow\infty}}{\sum_{{{i}={1}}}^{{n}}}{f{{\left({x}_{{i}}\right)}}}\Delta{x}\)
Here \(\displaystyle{f{{\left({x}_{{k}}\right)}}}=\frac{{1}}{{c}_{{k}}}\)
And p is a partition of \(\displaystyle{\left[{1},{4}\right]}\),
Therefore,
\(\displaystyle\lim_{{{n}\rightarrow\infty}}{\sum_{{{k}={1}}}^{{n}}}{\left(\frac{{1}}{{c}_{{k}}}\right)}\Delta{x}_{{k}}={\int_{{1}}^{{4}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\)
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