Question

use finite approximation to estimate the area under the graph of ( f(x)=x^{2} ) and above the graph of ( f(x)=0 ) from ( x_{0}=0 ) to ( x_{n}=12 ) using
i) a lower sum with two rectangles of equal width.
ii) a lower sum with four rectangles of equal width.
iii) an upper sum with two rectangles of equal width.
iv) an upper sum with four rectangles of equal width.

the estimated area using a lower 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

The interval is from \(x = 0\) to \(x=12\). For \(n = 2\) rectangles, the width \(\Delta x=\frac{12 - 0}{2}=6\).

Step2: Find the lower - sum

Since \(y = x^{2}\) is an increasing function on \([0,12]\), for a lower - sum with \(n = 2\) rectangles, we use the left - hand endpoints.
The left - hand endpoints are \(x_0=0\) and \(x_1 = 6\).
The area of the first rectangle \(A_1=f(x_0)\Delta x\), and since \(f(x)=x^{2}\), \(f(0)=0^{2}=0\).
The area of the second rectangle \(A_2=f(x_1)\Delta x\), \(f(6)=6^{2}=36\).
The lower - sum \(L_2=(f(0)+f(6))\Delta x=(0 + 36)\times6\).

Answer:

\(216\)