Question

use finite approximation to estimate the area under the graph of ( f(x)=7x^{2} ) and above the graph of ( g(x)=0 ) from ( x_{0}=0 ) to ( x_{n}=8 ) using
i) a right endpoint sum with two rectangles of equal width.
ii) a right endpoint sum with four rectangles of equal width.

the estimated area using a right endpoint sum with two rectangles of equal width is ( square ) square units
(simplify your answer. type an integer or a decimal.)

Explanation:

Step1: Calculate the width of each rectangle for two - rectangle case

The interval is from \(a = 0\) to \(b = 8\). For \(n = 2\) rectangles, the width \(\Delta x=\frac{b - a}{n}=\frac{8-0}{2}=4\).
The sub - intervals are \([0,4]\) and \([4,8]\). The right - endpoints are \(x_1 = 4\) and \(x_2=8\).

Step2: Calculate the right - endpoint sum

The right - endpoint sum formula is \(R_n=\sum_{i = 1}^{n}f(x_i)\Delta x\).
For \(n = 2\), \(R_2=f(4)\Delta x+f(8)\Delta x\).
Since \(f(x)=7x^{2}\), then \(f(4)=7\times4^{2}=7\times16 = 112\) and \(f(8)=7\times8^{2}=7\times64 = 448\).
\(R_2=(112 + 448)\times4\)
\(R_2=560\times4\)

Answer:

\(2240\)