Chapter 5.4, Problem 27ES

(a)

To determine

To calculate: The expression ${\displaystyle \sum _{k=1}^{4}f\left(x_k^{\ast }\right)\Delta x}$ as left endpoint of each subinterval, by first dividing the interval $\left[2,6\right]$ into $n=4$ subintervals of equal length, if $f\left(x\right)=3x+1$ . Illustrate it with a graph of $f$ that includes the rectangles whose areas are represented in the sum.

(b)

To determine

To calculate: The expression ${\displaystyle \sum _{k=1}^{4}f\left(x_k^{\ast }\right)\Delta x}$ as midpoint of each subinterval, by first dividing the interval $\left[2,6\right]$ into $n=4$ subintervals of equal length, if $f\left(x\right)=3x+1$ . Illustrate it with a graph of $f$ that includes the rectangles whose areas are represented in the sum.

(c)

To determine

To calculate: The expression ${\displaystyle \sum _{k=1}^{4}f\left(x_k^{\ast }\right)\Delta x}$ as right endpoint of each subinterval, by first dividing the interval $\left[2,6\right]$ into $n=4$ subintervals of equal length, if $f\left(x\right)=3x+1$ . Illustrate it with a graph of $f$ that includes the rectangles whose areas are represented in the sum.