Question

find the value v of the riemann sum $v = \sum_{k=1}^{n} f(c_k)\delta x_k$ for the function $f(x) = 2^x$ using the partition p = { 1, 2, 4, 6 }, where the $c_k$ are the right endpoints of the partition. v =

Explanation:

Step1: Calculate $\Delta x_k$ for each subinterval

First subinterval: $[1,2]$, $\Delta x_1 = 2-1=1$
Second subinterval: $[2,4]$, $\Delta x_2 = 4-2=2$
Third subinterval: $[4,6]$, $\Delta x_3 = 6-4=2$

Step2: Identify right endpoints $c_k$

$c_1=2$, $c_2=4$, $c_3=6$

Step3: Compute $f(c_k)\Delta x_k$ for each term

$f(c_1)\Delta x_1 = 2^2 \times 1 = 4 \times 1 = 4$
$f(c_2)\Delta x_2 = 2^4 \times 2 = 16 \times 2 = 32$
$f(c_3)\Delta x_3 = 2^6 \times 2 = 64 \times 2 = 128$

Step4: Sum all terms to get $V$

$V = 4 + 32 + 128$

Answer:

164