Express the limit as a deﬁnite integral.

-$\lim _{\mathrm||{P}|| \rightarrow 0} \sum_{k=1}^{n}\left(\sec ^{2} \mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}} $ where $\mathrm { P }$ is a partition of $[ - 5 \pi , 5 \pi ]$
A) $\int _ { - 5 \pi } ^ { 5 \pi } \tan x d x$
B) $\int _ { 5 \pi } ^ { - 5 \pi } \sec ^ { 2 } x d x$
C) $\int _ { 1 } ^ { n } \sec x d x$
D) $\int _ { - 5 \pi } ^ { 5 \pi } \sec ^ { 2 } x d x$

Question

Express the limit as a deﬁnite integral.

-$\lim _{\mathrm||{P}|| \rightarrow 0} \sum_{k=1}^{n}\left(\sec ^{2} \mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}} $ where $\mathrm { P }$ is a partition of $[ - 5 \pi , 5 \pi ]$
A) $\int _ { - 5 \pi } ^ { 5 \pi } \tan x d x$
B) $\int _ { 5 \pi } ^ { - 5 \pi } \sec ^ { 2 } x d x$
C) $\int _ { 1 } ^ { n } \sec x d x$
D) $\int _ { - 5 \pi } ^ { 5 \pi } \sec ^ { 2 } x d x$

Quizplus · Accepted Answer

We have
\begin{align*}
\lim_{||P||ightarrow 0} \sum_{k=1}^n \left(\sec^2 c_kight) \Delta x_k &=\lim_{||P||ightarrow 0} \sum_{k=1}^n \left(\sec^2 \frac{(k-1)5\pi}{n} ight) \frac{5\pi}{n}\
&=\int_{-5\pi}^{5\pi} \sec^2 x dx.
\end{align*}
Thus, the best choice is D, which is the same as the expression for the limit.

Express the Limit as a Deﬁnite Integral lim⁡∣∣P∣∣→0∑k=1n(sec⁡2ck)Δxk\lim _{\mathrm||{P}|| \rightarrow 0} \sum_{k=1}^{n}\left(\sec ^{2} \mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}} lim∣∣P∣∣→0​∑k=1n​(sec2ck​)Δxk​

Express the Limit as a Deﬁnite Integral $\lim _{\mathrm||{P}|| \rightarrow 0} \sum_{k=1}^{n}\left(\sec ^{2} \mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}}$