Single Variable Calculus, Chapter 5, 5.1, Section 5.1, Problem 4

Estimate the area under the graph of $f(x) = \sqrt{x}$ from $x = 0$ to $x = 4$ using four approximating rectangles at following sample points. Sketch the graph and the rectangles. Also, state that if your estimate is an underestimate or overestimate.

The width of the rectangle is..

$\displaystyle \Delta x = \frac{4 - 0}{4} = 1$

a.) Right endpoints $R_4$


$
\begin{equation}
\begin{aligned}

R_4 =& \sum \limits_{i = 1}^4 f(xi) \Delta x
\\
\\
R_4 =& 1 [f(1) + f(2) + f(3) + f(4)]
\\
\\
R_4 =& 1 [\sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4}]
\\
\\
R_4 =& 6.1463

\end{aligned}
\end{equation}
$








By using a sample point at right endpoints in an increasing function, we can say that this is an overestimate.

b.) Left endpoints $L_4$


$
\begin{equation}
\begin{aligned}

L_4 =& \sum \limits_{i = 1}^4 f(xi) \Delta x
\\
\\
L_4 =& 1 [f(0) + f(1) + f(2) + f(3) ]
\\
\\
L_4 =& [\sqrt{0} + \sqrt{1} + \sqrt{2} + \sqrt{3}]
\\
\\
L_4 =& 4.1463

\end{aligned}
\end{equation}
$








This time, we have an underestimate.

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