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 6000 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 four - rectangle case

The formula for the width of each rectangle is \(\Delta x=\frac{b - a}{n}\). Here, \(a = 0\), \(b = 20\), and \(n = 4\).
\(\Delta x=\frac{20-0}{4}=5\)

Step2: Determine the left - endpoint values

The sub - intervals are \([0,5]\), \([5,10]\), \([10,15]\), \([15,20]\). The left - endpoints are \(x_0 = 0\), \(x_1=5\), \(x_2 = 10\), \(x_3=15\)

Step3: Calculate the function values at the left - endpoints

For \(f(x)=6x^{2}\):

• When \(x = 0\), \(f(0)=6\times0^{2}=0\)
• When \(x = 5\), \(f(5)=6\times5^{2}=6\times25 = 150\)
• When \(x = 10\), \(f(10)=6\times10^{2}=6\times100 = 600\)
• When \(x = 15\), \(f(15)=6\times15^{2}=6\times225 = 1350\)

Step4: Calculate the left - endpoint sum

The left - endpoint sum \(L=\sum_{i = 0}^{n - 1}f(x_i)\Delta x\)
\(L=(f(0)+f(5)+f(10)+f(15))\times\Delta x\)
Substitute \(f(0) = 0\), \(f(5)=150\), \(f(10)=600\), \(f(15)=1350\) and \(\Delta x = 5\)
\(L=(0 + 150+600 + 1350)\times5=(2100)\times5=10500\)

Answer:

\(10500\)