Question

I need help question

Answer 1

`\begin{gathered} \Delta x=(b-a)/(n) \\ where, \\ b,a\text{ are the limits of integration and }n\text{ is the number of intervals} \\ \Delta x=(16-4)/(3)=4 \\ \\ \text{ Thus, we can find the intervals as follows:} \\ [4,4+4],[4+4,4+4+4],[4+4+4,4+4+4+4] \\ [4,8],[8,12],[12,16] \\ \\ \text{ Thus, the left handed x-values are:} \\ x\in[4,8,12] \\ \\ \text{ The right-handed x-values are:} \\ x\in[8,12,16] \\ \\ \text{ The function is:} \\ f(x)=3x^2+2x+6 \end{gathered}`

- The left-handed integral

`\begin{gathered} f(4)=3(4^2)+2(4)+6 \\ f(4)=62 \\ \\ f(8)=3(8^2)+2(8)+6 \\ f(8)=214 \\ \\ f(12)=3(12^2)+2(12)+6 \\ f(12)=462 \\ \\ \\ \text{ Thus, the sum is:} \\ L_3=\sum f(x)\Delta x=4\left(62+214+462\right)=2952. \end{gathered}`

The Right-handed integral

`\begin{gathered} f(8)=3(8^2)+2(8)+6 \\ f(8)=214 \\ \\ f(12)=3(12^2)+2(12)+6 \\ f(12)=462 \\ f(16)=806 \\ \\ \text{ Thus, the sum is:} \\ R_3=\sum f(x)\Delta x=4\left(214+462+806\right)=5928. \end{gathered}`

Answer

The answers are:

`\begin{gathered} L_3=2952 \\ \\ R_3=5928 \end{gathered}`