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 216 square units.
(simplify your answer. type an integer or a decimal.)

the estimated area using a lower sum with four rectangles of equal width is 378 square units.
(simplify your answer. type an integer or a decimal.)

the estimated area using an upper 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 sample points for the upper - sum

For an upper - sum with \(n = 2\) rectangles, the sample points are \(x_1=6\) and \(x_2 = 12\).

Step3: Calculate the sum

The upper - sum \(U=\sum_{i = 1}^{2}f(x_i)\Delta x\).
Since \(f(x)=x^{2}\), then \(f(6)=6^{2}=36\) and \(f(12)=12^{2}=144\).
\(U=(36 + 144)\times6\)
\(U = 180\times6\)
\(U=1080\)

Answer:

\(1080\)