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 (type an integer or a decimal.)

Explanation:

Step1: Calculate the width of each rectangle

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

Step2: Find the minimum value of the function on each sub - interval

First, find the derivative \(f(x)=3 - x^{2}+2x\), \(f^{\prime}(x)=-2x + 2\). Set \(f^{\prime}(x)=0\), then \(x = 1\).
For the sub - interval \([-1,1]\):
Evaluate \(f(x)\) at the endpoints. \(f(-1)=3-(-1)^{2}+2(-1)=3 - 1-2=0\), \(f(1)=3-1^{2}+2(1)=4\). The minimum value on \([-1,1]\) is \(f(-1)=0\).
For the sub - interval \([1,3]\):
\(f(1)=4\), \(f(3)=3-3^{2}+2(3)=3 - 9 + 6=0\). The minimum value on \([1,3]\) is \(f(3)=0\).

Step3: Calculate the lower sum

The lower sum \(L=\sum_{i = 1}^{n}f(x_{i})\Delta x\). Here \(n = 2\), \(\Delta x=2\), \(f(x_{1})=f(-1)=0\), \(f(x_{2})=f(3)=0\)
\(L=(f(-1)+f(3))\Delta x=(0 + 0)\times2=0\)

Answer:

\(0\)