Let f (x) = $\frac { 1 } { x }$ on the interval [1,2]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i }$ if each $x _ { i } ^ { * }$ is the left endpoint of its subinterval.
A) $\frac { 319 } { 105 }$

B) $\frac { 319 } { 420 }$
C) $\frac { 533 } { 210 }$
D) $\frac { 533 } { 840 }$
E) $\frac { 11 } { 2 }$
F) $\frac { 11 } { 8 }$

G) $\frac { 13 } { 2 }$

H) $\frac { 13 } { 8 }$

Question

Let f (x) = $\frac { 1 } { x }$ on the interval [1,2]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i }$ if each $x _ { i } ^ { * }$ is the left endpoint of its subinterval.
A) $\frac { 319 } { 105 }$

B) $\frac { 319 } { 420 }$ 
C) $\frac { 533 } { 210 }$ 
D) $\frac { 533 } { 840 }$ 
E) $\frac { 11 } { 2 }$ 
F) $\frac { 11 } { 8 }$

G) $\frac { 13 } { 2 }$

H) $\frac { 13 } { 8 }$

Quizplus · Accepted Answer

The interval [1,2] is divided into four equal subintervals, each of length $\Delta x = \frac{2-1}{4} = \frac{1}{4}$. The left endpoints are $x_1^* = 1$, $x_2^* = 1.25$, $x_3^* = 1.5$, and $x_4^* = 1.75$. The Riemann sum is calculated as $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x = \left( \frac{1}{1} + \frac{1}{1.25} + \frac{1}{1.5} + \frac{1}{1.75} \right) \frac{1}{4} = \left( 1 + \frac{4}{5} + \frac{2}{3} + \frac{4}{7} \right) \frac{1}{4} = \frac{319}{420}$.

Let F (X) = 1x\frac { 1 } { x }x1​

Let F (X) = $\frac { 1 } { x }$