Question

use finite approximations to estimate the area under the graph of the function ( f(x)=3 - x^{2}+2x ) between ( x=-1 ) and ( x = 3 ) for each of the following cases.
a. using a lower sum with two rectangles of equal width
b. using a lower sum with four rectangles of equal width
c. using an upper sum with two rectangles of equal width
d. using an upper sum with four rectangles of equal width
a. the area under the graph of ( f(x) ) using a lower sum with two rectangles of equal width is 0 (type an integer or a decimal.)
b. the area under the graph of ( f(x) ) using a lower sum with four rectangles of equal width is 6 (type an integer or a decimal.)
c. the area under the graph of ( f(x) ) using an upper sum with two rectangles of equal width is (type an integer or a decimal.)

Explanation:

Step1: Calculate the width of each rectangle

The interval is from \(x = - 1\) to \(x = 3\). The length of the interval is \(b - a=3-(-1) = 4\).
For \(n = 2\) rectangles, \(\Delta x=\frac{b - a}{n}=\frac{4}{2}=2\).
The sub - intervals are \([-1,1]\) and \([1,3]\).

Step2: Find the upper sum

For an upper sum, we evaluate the function at the right - hand endpoints of each sub - interval.
For the first sub - interval \([-1,1]\), the right - hand endpoint is \(x = 1\). \(f(1)=3-1^{2}+2\times1=3 - 1+2=4\).
For the second sub - interval \([1,3]\), the right - hand endpoint is \(x = 3\). \(f(3)=3-3^{2}+2\times3=3 - 9 + 6=0\).
The upper sum \(U=\sum_{i = 1}^{n}f(x_{i})\Delta x\). Here \(n = 2\), \(\Delta x=2\), \(U=f(1)\Delta x+f(3)\Delta x\).
Substitute the values: \(U=(4 + 0)\times2=8\).

Answer:

\(8\)