Question

# Express the limits as definite integrals. \lim_{||P||\rightarrow 0}\sum_{k=1}^{n}(\frac{1}{C_{k}})\triangle x_{k}, where P is a partition of [1,4]

Applications of integrals

Express the limits as definite integrals. $$\displaystyle\lim_{{{P}\rightarrow{0}}}{\sum_{{{k}={1}}}^{{{n}}}}{\left({\frac{{{1}}}{{{C}_{{{k}}}}}}\right)}\triangle{x}_{{{k}}}$$, where P is a partition of [1,4]

2021-05-18
Step 1
Given
$$\lim_{||P||\rightarrow 0}\sum_{k=1}^{n}(\frac{1}{C_{k}})\triangle x_{k}$$
Step 2
To express limits as a definite integrals.
The definition of definite integral is,
$$\int_{a}^{b}f(x)dx=\lim_{n\rightarrow \infty}\sum_{i=1}^{n}f(x_{i})\triangle x$$
Here $$f(x_{k})=\frac{1}{C_{k}}$$
And p is a partition of [1,4],
Therefore,
$$\lim_{n\rightarrow \infty}\sum_{k=1}^{n}(\frac{1}{C_{k}})\triangle x_{k}=\int_{1}^{4}f(x)dx$$