Express the limit as a deﬁnite integral.
- $\text { where } \mathrm{P} \text { is a partition of }[-\pi / 3,0]$
A) $\int _ { - \pi / 3 } ^ { 0 } \sin x d x$
B) $\int _ { 0 } ^ { \pi / 3 } \sin x d x$
C) $\int _ { 1 } ^ { n } \sin x d x$
D) $\int _ { - \pi / 3 } ^ { 0 } \cos x d x$

Question

Express the limit as a deﬁnite integral.
-  $\text { where } \mathrm{P} \text { is a partition of }[-\pi / 3,0]$ 
A) $\int _ { - \pi / 3 } ^ { 0 } \sin x d x$ 
B) $\int _ { 0 } ^ { \pi / 3 } \sin x d x$ 
C) $\int _ { 1 } ^ { n } \sin x d x$ 
D) $\int _ { - \pi / 3 } ^ { 0 } \cos x d x$

Quizplus · Accepted Answer

The limit can be expressed as:

$$\lim_{||P||	o 0} \sum_{i=1}^n \sin(x_i^*) \Delta x_i $$

where $x_i^*$ is any point in the $i$th subinterval of the partition $P$, and $\Delta x_i$ is the length of the $i$th subinterval.

Since the interval $[-\pi/3,0]$ is on the negative $x$-axis, the function $\sin x$ is negative on this interval. Therefore, we can write:

$$\lim_{||P||	o 0} \sum_{i=1}^n \sin(x_i^*) \Delta x_i = -\int_{-\pi/3}^0 \sin x dx$$

Thus, the limit can be expressed as the definite integral $\int_{-\pi/3}^0 \sin x dx$.

Express the Limit as a Deﬁnite Integral where P is a partition of [−π/3,0]\text { where } \mathrm{P} \text { is a partition of }[-\pi / 3,0] where P is a partition of [−π/3,0]

Express the Limit as a Deﬁnite Integral $\text { where } \mathrm{P} \text { is a partition of }[-\pi / 3,0]$