Question

consider the integral
int_{3}^{12}left(4 x^{2}+4 x+4
ight) d x
(a) find the riemann sum for this integral using left endpoints and ( n = 3 ).
( l_{3}= )
(b) find the riemann sum for this same integral, using right endpoints and ( n = 3 ).
( r_{3}= )

Explanation:

Step1: Calculate the width of each sub - interval

The formula for the width of a sub - interval is \(\Delta x=\frac{b - a}{n}\). Here, \(a = 3\), \(b = 12\), and \(n = 3\).
\(\Delta x=\frac{12 - 3}{3}=\frac{9}{3}=3\)

Step2: Find the left endpoints

The left endpoints \(x_i\) for \(i = 0,1,2\) are \(x_0=a=3\), \(x_1=a+\Delta x=3 + 3=6\), \(x_2=a + 2\Delta x=3+2\times3 = 9\)

Step3: Calculate the Riemann sum using left endpoints (\(L_3\))

The Riemann sum formula for left endpoints is \(L_n=\sum_{i = 0}^{n-1}f(x_i)\Delta x\).
\(f(x)=4x^{2}+4x + 4\)
\(f(3)=4\times3^{2}+4\times3 + 4=4\times9+12 + 4=36+12 + 4=52\)
\(f(6)=4\times6^{2}+4\times6 + 4=4\times36+24 + 4=144+24 + 4=172\)
\(f(9)=4\times9^{2}+4\times9 + 4=4\times81+36 + 4=324+36 + 4=364\)
\(L_3=(f(3)+f(6)+f(9))\Delta x=(52 + 172+364)\times3=(588)\times3 = 1764\)

Step4: Find the right endpoints

The right endpoints \(x_i\) for \(i = 1,2,3\) are \(x_1=a+\Delta x=3 + 3=6\), \(x_2=a + 2\Delta x=3+2\times3 = 9\), \(x_3=a+3\Delta x=3 + 3\times3=12\)

Step5: Calculate the Riemann sum using right endpoints (\(R_3\))

\(f(6)=4\times6^{2}+4\times6 + 4=172\)
\(f(9)=4\times9^{2}+4\times9 + 4=364\)
\(f(12)=4\times12^{2}+4\times12 + 4=4\times144+48 + 4=576+48 + 4=628\)
\(R_3=(f(6)+f(9)+f(12))\Delta x=(172+364 + 628)\times3=(1164)\times3=3492\)

Answer:

a. \(L_3 = 1764\)
b. \(R_3=3492\)