Use a ﬁnite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$f ( x ) = \frac { 1 } { x }$ between $x = 1$ and $x = 7$ using the "midpoint rule" with two rectangles of equal width.
A) $\frac { 48 } { 55 }$
B) $\frac { 96 } { 55 }$
C) $\frac { 876 } { 3025 }$
D) $\frac { 584 } { 3025 }$

Question

Quizplus · Accepted Answer

To use the midpoint rule with two rectangles of equal width to estimate the area under $f(x) = \frac{1}{x}$ from $x = 1$ to $x = 7$, we first divide the interval into two equal parts. The total width is $7 - 1 = 6$, so each rectangle has a width of $3$. The midpoints for these intervals are $x = 2.5$ for the first rectangle (midpoint between 1 and 4) and $x = 5.5$ for the second rectangle (midpoint between 4 and 7). The height of each rectangle is given by the value of the function at these midpoints: $f(2.5) = \frac{1}{2.5}$ and $f(5.5) = \frac{1}{5.5}$. The area is then approximated by the sum of the areas of these rectangles:Area ≈ $\left(\frac{1}{2.5} \times 3\right) + \left(\frac{1}{5.5} \times 3\right) = \frac{3}{2.5} + \frac{3}{5.5} = \frac{6}{5} + \frac{6}{11} = \frac{66 + 30}{55} = \frac{96}{55}$.

Use a ﬁNite Approximation to Estimate the Area Under the Graph