use finite approximation to estimate the area under the graph of f(x)=6x² and above the graph of f(x)=0 from…

use finite approximation to estimate the area under the graph of f(x)=6x² and above the graph of f(x)=0 from x₀ = 0 to xₙ = 6 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 units. (simplify your answer. type an integer or a decimal.)

use finite approximation to estimate the area under the graph of f(x)=6x² and above the graph of f(x)=0 from x₀ = 0 to xₙ = 6 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 units. (simplify your answer. type an integer or a decimal.)

Answer

Explanation:

Step1: Calculate width of rectangles

The interval is from $x_0 = 0$ to $x_n=6$. For $n = 2$ rectangles, the width $\Delta x=\frac{6 - 0}{2}=3$.

Step2: Determine left - hand endpoints for lower sum

The left - hand endpoints of the two sub - intervals $[0,3]$ and $[3,6]$ are $x_1 = 0$ and $x_2=3$.

Step3: Calculate the function values at endpoints

$f(x)=6x^{2}$, so $f(0)=6\times0^{2}=0$ and $f(3)=6\times3^{2}=6\times9 = 54$.

Step4: Calculate the lower sum

The lower sum $L_2=\sum_{i = 1}^{2}f(x_{i})\Delta x=f(0)\times3+f(3)\times3=(0 + 54)\times3=162$.

Answer:

162