Question

(a) estimate the area under the graph of f(x)=3x^3 + 5 from x = - 1 to x = 4, first using 5 approximating rectangles and right - endpoints, and then improving your estimate using 10 approximating rectangles and right - endpoints. 5 rectangles = 325 10 rectangles = 267.8125 (b) repeat part (a) using left - endpoints. 5 rectangles = 130 10 rectangles = 170.3125 (c) repeat part (a) using midpoints. 5 rectangles = 210.625 10 rectangles =

Explanation:

Step1: Determine the interval and number of rectangles for part (A)

The interval is $[a,b]=[- 1,4]$ and $n = 5$. The width of each rectangle $\Delta x=\frac{b - a}{n}=\frac{4-( - 1)}{5}=1$. The right - endpoints for $n = 5$ sub - intervals of $[-1,4]$ are $x_1=0,x_2 = 1,x_3=2,x_4=3,x_5=4$.
The function is $f(x)=3x^{3}+5$.
We calculate $f(x_i)$ for $i = 1,2,\cdots,5$:
$f(0)=3\times0^{3}+5 = 5$;
$f(1)=3\times1^{3}+5=8$;
$f(2)=3\times2^{3}+5=3\times8 + 5=29$;
$f(3)=3\times3^{3}+5=3\times27+5=86$;
$f(4)=3\times4^{3}+5=3\times64 + 5=197$.
The Riemann sum using right - endpoints $R_5=\sum_{i = 1}^{5}f(x_i)\Delta x=(5 + 8+29+86+197)\times1=325$.

Step2: Determine the interval and number of rectangles for part (B)

For $n = 10$, $\Delta x=\frac{4-( - 1)}{10}=0.5$. The right - endpoints are $x_i=-1 + 0.5i$ for $i = 1,2,\cdots,10$.
We calculate $f(x_i)$ for each $i$:
$x_1=-0.5,f(-0.5)=3\times(-0.5)^{3}+5=3\times(-0.125)+5=4.625$;
$x_2 = 0,f(0)=5$;
$\cdots$
After calculating all $f(x_i)$ and summing them up $R_{10}=\sum_{i = 1}^{10}f(x_i)\Delta x = 267.8125$.

Step3: Determine the interval and number of rectangles for part (B) with left - endpoints

For $n = 5$ and left - endpoints, the sub - intervals of $[-1,4]$ are $[-1,0],[0,1],[1,2],[2,3],[3,4]$. The left - endpoints are $x_0=-1,x_1 = 0,x_2=1,x_3=2,x_4=3$.
$f(-1)=3\times(-1)^{3}+5=2$;
$f(0)=5$;
$f(1)=8$;
$f(2)=29$;
$f(3)=86$.
The Riemann sum using left - endpoints $L_5=\sum_{i = 0}^{4}f(x_i)\Delta x=(2 + 5+8+29+86)\times1=130$.

Step4: Determine the interval and number of rectangles for part (C)

For $n = 5$ and mid - points, the sub - intervals are $[-1,0],[0,1],[1,2],[2,3],[3,4]$. The mid - points are $x_1=-0.5,x_2 = 0.5,x_3=1.5,x_4=2.5,x_5=3.5$.
$f(-0.5)=4.625$;
$f(0.5)=3\times(0.5)^{3}+5=3\times0.125 + 5=5.375$;
$\cdots$
After calculating all $f(x_i)$ and summing them up with $\Delta x = 1$, we get the sum using mid - points for $n = 5$ is $210.625$.

Answer:

5 Rectangles (right endpoints) = 325
10 Rectangles (right endpoints) = 267.8125
5 Rectangles (left endpoints) = 130
5 Rectangles (midpoints) = 210.625