The sum of the areas for the right-endpoint rectangles of the function$A(x)=\sqrt{2x+1}$ using $6$ rectangles on the interval $1\le x\le4$ is $\left(0.5\right)[\left(2 + 2.236 + … + 3\right)] = 7.579$. 

1. What does $0.5$ represent?

   Answer (a):

   Width

   Graph (a):

   

2. Let $b\left(x\right)=2+A\left(x\right)$. Use what you know to find the sum of the areas of the $6$ right-endpoint rectangles of $b\left(x\right)$ on $1 ≤ x ≤ 4$.

   Answer (b):

   The $2$ in the express $2 + A\left(x\right)$ raises the function up $2$. So the total area is increased by $6$ because the base was $3$.

3. Find the sum of the $6$ right-endpoint rectangles for $c\left(x\right) = 1000 + A\left(x\right)$ for $1 ≤ x ≤ 4$.

   Hint (c):

   Since the whole graph is raised up $1000$ and the base is still $3$, how much more area would the new function have than $A\left(x\right)$?