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

Explanation:

Step1: Calculate the width of each rectangle

For \(n = 4\) rectangles, the width \(\Delta x=\frac{12 - 0}{4}=3\). The sub - intervals are \([0,3]\), \([3,6]\), \([6,9]\), \([9,12]\).

Step2: Find the lower sum

Since \(y = x^{2}\) is an increasing function on \([0,12]\), for the lower sum, we use the left - hand endpoints of each sub - interval.
The left - hand endpoints are \(x_0 = 0\), \(x_1=3\), \(x_2 = 6\), \(x_3=9\).
The lower sum \(L=\sum_{i = 0}^{3}f(x_i)\Delta x\)

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Answer:

\(378\)