Express the limit as a deﬁnite integral.

-$\lim_{\|\mathrm{P}\| \rightarrow 0} \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{5c}{\mathrm{k}^{5} \Delta \mathrm{x}_{\mathrm{k}^{\prime}}} $ where $\mathrm { P }$ is a partition of $[ 5,12 ]$
A) $\int _ { 12 } ^ { 5 } 5 x ^ { 5 } d x$
B) $\int _ { 1 } ^ { n } 5 x d x$
C) $\int _ { 5 } ^ { 12 } 25 x ^ { 4 } d x$
D) $\int _ { 5 } ^ { 12 } 5 x ^ { 5 } d x$

Question

Express the limit as a deﬁnite integral.

-$\lim_{\|\mathrm{P}\| \rightarrow 0} \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{5c}{\mathrm{k}^{5} \Delta \mathrm{x}_{\mathrm{k}^{\prime}}} $ where $\mathrm { P }$ is a partition of $[ 5,12 ]$
A) $\int _ { 12 } ^ { 5 } 5 x ^ { 5 } d x$
B) $\int _ { 1 } ^ { n } 5 x d x$
C) $\int _ { 5 } ^ { 12 } 25 x ^ { 4 } d x$
D) $\int _ { 5 } ^ { 12 } 5 x ^ { 5 } d x$

Quizplus · Accepted Answer

The limit can be expressed as:
$$\lim_{\|\mathrm{P}\| \rightarrow 0} \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{5c}{\mathrm{k}^{5} \Delta \mathrm{x}_{\mathrm{k}^{\prime}}} = \int_5^{12} 5x^5 dx$$
This is because as the partition size approaches 0, the sum approaches the definite integral of the function. Therefore, the best choice is D.

Express the Limit as a Deﬁnite Integral lim⁡∥P∥→0∑k=1n5ck5Δxk′\lim_{\|\mathrm{P}\| \rightarrow 0} \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{5c}{\mathrm{k}^{5} \Delta \mathrm{x}_{\mathrm{k}^{\prime}}} lim∥P∥→0​∑k=1n​5ck5Δxk′​

Express the Limit as a Deﬁnite Integral $\lim_{\|\mathrm{P}\| \rightarrow 0} \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{5c}{\mathrm{k}^{5} \Delta \mathrm{x}_{\mathrm{k}^{\prime}}}$