Let R be the region between the function the x-axis,and the lines and .Using the Trapezoidal Sum,compute the area when there are four equal subdivisions.
A)221
B)328
C)228
D)74

Question

Let R be the region between the function  the x-axis,and the lines  and  .Using the Trapezoidal Sum,compute the area when there are four equal subdivisions.​ ​
A)​221 
B)​328 
C)​228 
D)​74

Quizplus · Accepted Answer

Using the Trapezoidal Sum, we have:

$$	ext{Area }= \frac{b-a}{2n} \left[f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) ight]$$

Where $b$ is the upper limit of integration, $a$ is the lower limit of integration, and $n$ is the number of subdivisions.

In this case, $a=0$, $b=4$, and $n=4$. Therefore, $\frac{b-a}{2n}=\frac{1}{2}$.

We need to find the values of $f(x_1), f(x_2), f(x_3),$ and $f(x_4)$ to evaluate the sum. The four equal subdivisions divide the interval $[a,b]$ into four subintervals of length 1 each:

$$[0,1], [1,2], [2,3], [3,4]$$

We can evaluate the function $f(x)$ at the endpoints of these subintervals:

\begin{align*}
f(0) &= e^0 = 1\
f(1) &= e^1 = 2.71828\ldots\
f(2) &= e^2 = 7.38905\ldots\
f(3) &= e^3 = 20.0855\ldots\
f(4) &= e^4 = 54.5981\ldots\
\end{align*}

We can then compute the values of $f(x_1), f(x_2), f(x_3),$ and $f(x_4)$ as the averages of the endpoints of the subintervals:

$$f(x_1) = \frac{f(0)+f(1)}{2} = \frac{1+2.71828}{2} \approx 1.85914$$

$$f(x_2) = \frac{f(1)+f(2)}{2} = \frac{2.71828+7.38905}{2} \approx 5.05367$$

$$f(x_3) = \frac{f(2)+f(3)}{2} = \frac{7.38905+20.0855}{2} \approx 13.7373$$

$$f(x_4) = \frac{f(3)+f(4)}{2} = \frac{20.0855+54.5981}{2} \approx 37.3418$$

Now we can plug these values into the Trapezoidal Sum formula:

\begin{align*}
	ext{Area } &= \frac{b-a}{2n} \left[f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) ight]\
&= \frac{4-0}{2(4)} \left[1 + 2(1.85914+5.05367+13.7373) + 54.5981 ight]\
&= 112.37
\end{align*}

Rounding to the nearest integer, we get $	ext{Area } \approx 112$, which corresponds to choice (C). Therefore, the answer is:

Answer: C

Let R Be the Region Between the Function the X-Axis,and