# 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]

Express the limits as definite integral.

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Roosevelt Houghton
Step 1
Given
$\underset{‖p\to 0‖}{lim}\sum _{k=1}^{n}\left(\frac{1}{{c}_{k}}\right)\mathrm{\Delta }{x}_{k}$
Step 2
To express limits as a definite integrals.
The definition of definite integral is,
${\int }_{a}^{b}f\left(x\right)dx=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}f\left({x}_{i}\right)\mathrm{\Delta }x$
Here $f\left({x}_{k}\right)=\frac{1}{{c}_{k}}$
And p is a partition of $\left[1,4\right]$,
Therefore,
$\underset{n\to \mathrm{\infty }}{lim}\sum _{k=1}^{n}\left(\frac{1}{{c}_{k}}\right)\mathrm{\Delta }{x}_{k}={\int }_{1}^{4}f\left(x\right)dx$