Express the limit as a deﬁnite integral.

-$\lim _{||P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sec c_{k} \tan c_{k}\right) \Delta x_{k^{\prime}} $, where $\mathrm { P }$ is a partition of $[ 0,4 \pi ]$
A) $\int _ { 4 \pi } ^ { 0 } ( \sec x \tan x ) d x$
B) $\int _ { 0 } ^ { 4 \pi } \sec x d x$
C) $\int _ { 0 } ^ { 4 \pi } ( \sec x \tan x ) d x$
D) $\int _ { 1 } ^ { n } \sec x d x$

Question

Express the limit as a deﬁnite integral.

-$\lim _{||P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sec c_{k} \tan c_{k}\right) \Delta x_{k^{\prime}} $, where $\mathrm { P }$ is a partition of $[ 0,4 \pi ]$
A) $\int _ { 4 \pi } ^ { 0 } ( \sec x \tan x ) d x$
B) $\int _ { 0 } ^ { 4 \pi } \sec x d x$
C) $\int _ { 0 } ^ { 4 \pi } ( \sec x \tan x ) d x$
D) $\int _ { 1 } ^ { n } \sec x d x$

Quizplus · Accepted Answer

The limit given is in the form of a Riemann sum, which can be expressed as a definite integral. Using the definition of definite integrals, we can express the limit as:
$$\lim _{||P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sec c_{k} \tan c_{k}\right) \Delta x_{k^{\prime}} =\int_0^{4\pi}\sec x \tan x dx$$
Therefore, the correct choice is C.

Express the Limit as a Deﬁnite Integral lim⁡∣∣P∥→0∑k=1n(sec⁡cktan⁡ck)Δxk′\lim _{||P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sec c_{k} \tan c_{k}\right) \Delta x_{k^{\prime}} lim∣∣P∥→0​∑k=1n​(secck​tanck​)Δxk′​

Express the Limit as a Deﬁnite Integral $\lim _{||P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sec c_{k} \tan c_{k}\right) \Delta x_{k^{\prime}}$