(a) estimate the area under the graph of f(x)=3x^3 + 5 from x = - 1 to x = 4, first using 5 approximating…

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