Question

use finite approximations to estimate the area under the graph of the function $f(x)=8-x^{2}+2x$ between $x=-2$ and $x=4$ 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

Explanation:

First, we define the interval $[a,b] = [-2,4]$, function $f(x)=8-x^2+2x = -(x^2-2x-8) = -(x-4)(x+2)$. This is a downward-opening parabola with vertex at $x = -\frac{b}{2a} = 1$, $f(1)=8-1+2=9$.

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Part a: Lower sum, 2 rectangles

Step1: Find rectangle width

$\Delta x = \frac{4 - (-2)}{2} = 3$

Step2: Identify subintervals

Subintervals: $[-2,1], [1,4]$

Step3: Find min $f(x)$ on each interval

For $[-2,1]$ (increasing interval): min at $x=-2$, $f(-2)=8-4-4=0$
For $[1,4]$ (decreasing interval): min at $x=4$, $f(4)=8-16+8=0$

Step4: Calculate lower sum

$\text{Lower Sum} = \Delta x [f(-2) + f(4)] = 3(0+0)=0$

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Part b: Lower sum, 4 rectangles

Step1: Find rectangle width

$\Delta x = \frac{4 - (-2)}{4} = 1.5$

Step2: Identify subintervals

Subintervals: $[-2,-0.5], [-0.5,1], [1,2.5], [2.5,4]$

Step3: Find min $f(x)$ on each interval

• $[-2,-0.5]$ (increasing): min at $x=-2$, $f(-2)=0$
• $[-0.5,1]$ (increasing): min at $x=-0.5$, $f(-0.5)=8-0.25-1=6.75$
• $[1,2.5]$ (decreasing): min at $x=2.5$, $f(2.5)=8-6.25+5=6.75$
• $[2.5,4]$ (decreasing): min at $x=4$, $f(4)=0$

Step4: Calculate lower sum

$\text{Lower Sum} = 1.5(0 + 6.75 + 6.75 + 0) = 1.5(13.5)=20.25$

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Part c: Upper sum, 2 rectangles

Step1: Find rectangle width

$\Delta x = \frac{4 - (-2)}{2} = 3$

Step2: Identify subintervals

Subintervals: $[-2,1], [1,4]$

Step3: Find max $f(x)$ on each interval

For $[-2,1]$ (increasing): max at $x=1$, $f(1)=9$
For $[1,4]$ (decreasing): max at $x=1$, $f(1)=9$

Step4: Calculate upper sum

$\text{Upper Sum} = 3(9+9)=54$

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Part d: Upper sum, 4 rectangles

Step1: Find rectangle width

$\Delta x = \frac{4 - (-2)}{4} = 1.5$

Step2: Identify subintervals

Subintervals: $[-2,-0.5], [-0.5,1], [1,2.5], [2.5,4]$

Step3: Find max $f(x)$ on each interval

• $[-2,-0.5]$ (increasing): max at $x=-0.5$, $f(-0.5)=6.75$
• $[-0.5,1]$ (increasing): max at $x=1$, $f(1)=9$
• $[1,2.5]$ (decreasing): max at $x=1$, $f(1)=9$
• $[2.5,4]$ (decreasing): max at $x=2.5$, $f(2.5)=6.75$

Step4: Calculate upper sum

$\text{Upper Sum} = 1.5(6.75 + 9 + 9 + 6.75) = 1.5(31.5)=47.25$

Answer:

a. $\boldsymbol{0}$
b. $\boldsymbol{20.25}$
c. $\boldsymbol{54}$
d. $\boldsymbol{47.25}$