Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$\mathrm { f } ( \mathrm { x } ) = 5 \mathrm { x } \sin \pi \mathrm { x }$ on $[ 0,4 ]$ divided into 4 subintervals
A) $- \frac { 5 } { 4 }$
B) 0
C) $- \frac { 5 } { 2 }$
D) $- 10$

Question

Quizplus · Accepted Answer

We have to use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals.

The interval [0,4] is divided into 4 subintervals: [0,1], [1,2], [2,3], [3,4]

The length of each subinterval is:

$$\Delta x = \frac{4-0}{4} = 1$$

The midpoints of each subinterval are:

$$x_1 = \frac{1}{2}, x_2 = \frac{3}{2}, x_3 = \frac{5}{2}, x_4 = \frac{7}{2}$$

The sum we need to evaluate is:

$$\frac{1}{4} \sum_{i=1}^{4} f(x_i) \Delta x$$

where

$$f(x) = 5x\sin(\pi x)$$

So,

\begin{align*}
\frac{1}{4} \sum_{i=1}^{4} f(x_i) \Delta x &= \frac{1}{4}[(5\cdot \frac{1}{2}\sin(\pi \cdot \frac{1}{2})) + (5\cdot \frac{3}{2}\sin(\pi \cdot \frac{3}{2}))\
&\quad+ (5\cdot \frac{5}{2}\sin(\pi \cdot \frac{5}{2})) + (5\cdot \frac{7}{2}\sin(\pi \cdot \frac{7}{2}))] \cdot 1\
&= \frac{5}{4}[\sin(\frac{\pi}{2}) + 3\sin(\frac{3\pi}{2}) + 5\sin(\frac{5\pi}{2}) + 7\sin(\frac{7\pi}{2})]\
&= \frac{5}{4}[-1 + 3(-1) + 5(1) + 7(-1)]\
&= \frac{5}{4}(-1 - 3 + 5 - 7)\
&= -\frac{5}{2}
\end{align*}

Therefore, the best choice is (C), i.e., $-\frac{5}{2}$.

Use a ﬁNite Sum to Estimate the Average Value of the Function