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2-158.

The area under the function $f(x)=3x^3+1$ on the interval from $1$ to $3$ is to be approximated using $10$ left-endpoint rectangles.

1. Write the summation notation that will represent the area.

Sketch the function.

Draw $10$ rectangles between $x = 1$ and $3$.

What is the width for each rectangle? What are the $x$-values for the heights?

$\frac{3-1}{10}=\frac{2}{10}=0.2$

$x_{0} = 1$; $x_{1} = 1.2$; $x_{2} = 1.4$; $x_{3} = 1.6$; $\dots$; $x_{9} = 2.8$

What expression needs to go into $f(x)$ to convert the integer values to $1$, $1.2$, $1.4$, $1.6$, $\dots$, $2.8$?

$0.2k+1$

Write the summation now. Be careful with your $k$-values.

2. Make a small change to your answer in part (a) so that the sum will find the area using right-endpoint rectangles.

What $k$-values should you start and stop with?

3. Make a small change to your answer in part (a) so that the sum will find the area using midpoint rectangles.

Change the inside function to: $0.2k + 1.1$ so that it is starting at the middle of each rectangle.