(1 point)
We want to find the area of a region S which lies below the graph of $f(x) = 4x + 3$ and above the interval $[a, b] = [5, 10]$ on the x-axis. Thus
$S = \{(x, y) : 5 \le x \le 10, 0 \le y \le 4x + 3\}$.
To do this we begin by dividing the interval $[5, 10]$ into $N$ equal subintervals using the points $x_0 = 5, x_1, x_2, \dots, x_N = 10$.
The length of each sub-interval $[x_{k-1}, x_k]$ is $\frac{5}{N}$
Find a formula for the $k^{th}$ division point in terms of $k$ and $N$.
$x_k = 5 + \frac{5k}{N}$
help (formulas)
To approximate the area, we construct rectangles as the one pictured above. The base is the interval $[x_{k-1}, x_k]$ on the x-axis and the height is $f(x_k)$. Find a formula for $A_k$, the area of
this rectangle.
$A_k = $
Find an expression of the sum of the areas $A_k$.
$S_N = \sum_{k=1}^N A_k$
It will be helpful to use the formulas for the sums of powers of integers from the textbook.
Take the limit as $N$ tends to infinity to find the area of $S$
$S = \lim_{N \to \infty} S_N = $
