Question

use finite approximation to estimate the area under the graph of ( f(x)=6x^{2} ) and above the graph of ( g(x)=0 ) from ( x_{0}=0 ) to ( x_{n}=20 ) using
i) a left endpoint sum with two rectangles of equal width.
ii) a left endpoint 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 formula for the width \(\Delta x=\frac{b - a}{n}\). Here, \(a = 0\), \(b = 20\), and \(n = 2\).
\(\Delta x=\frac{20-0}{2}=10\)

Step2: Find the left - endpoints

The left - endpoints are \(x_0=a = 0\) and \(x_1=a+\Delta x=0 + 10=10\)

Step3: Calculate the function values at the left - endpoints

For \(x = 0\), \(f(0)=6\times0^{2}=0\)
For \(x = 10\), \(f(10)=6\times10^{2}=6\times100 = 600\)

Step4: Calculate the left - endpoint sum

The left - endpoint sum \(L=\sum_{i = 0}^{n-1}f(x_i)\Delta x\)
\(L=(f(0)+f(10))\Delta x=(0 + 600)\times10=6000\)

Answer:

\(6000\)