approximate the area of the region bounded by $y = -\\frac{1}{4}x^{2}+9…
$\frac{26 + 34}{2} = 30$
Question
approximate the area of the region bounded by $y = -\\frac{1}{4}x^{2}+9$, the x-axis, $x = -4$, and $x = 0$ by finding the combined area of the rectangles (as shown in each figure) and averaging the results.
Solution Steps
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Understand the question
approximate the area of the region bounded by $y = -\\frac{1}{4}x^{2}+9$, the x-axis, $x = -4$, and $x = 0$ by finding the combined area of the rectangles (as shown in each figure) and averaging the results.
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Explanation
Step1: Identify rectangle widths
The interval from $x=-4$ to $x=0$ is split into 2 equal subintervals, so width $\Delta x = \frac{0 - (-4)}{2} = 2$.
Step2: Calculate left rectangle area
Left rectangles use heights at $x=-4$ and $x=-2$:
Height at $x=-4$: $y(-4) = -\frac{1}{4}(-4)^2 + 9 = 5$
Height at $x=-2$: $y(-2) = -\frac{1}{4}(-2)^2 + 9 = 8$
Area $A_{\text{left}} = \Delta x \cdot (y(-4) + y(-2)) = 2 \cdot (5 + 8) = 26$Step3: Calculate right rectangle area
Right rectangles use heights at $x=-2$ and $x=0$:
Height at $x=-2$: $y(-2) = 8$
Height at $x=0$: $y(0) = -\frac{1}{4}(0)^2 + 9 = 9$
Area $A_{\text{right}} = \Delta x \cdot (y(-2) + y(0)) = 2 \cdot (8 + 9) = 34$Step4: Average the two areas
Find the mean of the two areas:
$\text{Average Area} = \frac{A_{\text{left}} + A_{\text{right}}}{2}$ -
Final answer
$\frac{26 + 34}{2} = 30$
Answer
Explanation
Step1: Identify rectangle widths
The interval from $x=-4$ to $x=0$ is split into 2 equal subintervals, so width $\Delta x = \frac{0 - (-4)}{2} = 2$.
Step2: Calculate left rectangle area
Left rectangles use heights at $x=-4$ and $x=-2$:
Height at $x=-4$: $y(-4) = -\frac{1}{4}(-4)^2 + 9 = 5$
Height at $x=-2$: $y(-2) = -\frac{1}{4}(-2)^2 + 9 = 8$
Area $A_{\text{left}} = \Delta x \cdot (y(-4) + y(-2)) = 2 \cdot (5 + 8) = 26$
Step3: Calculate right rectangle area
Right rectangles use heights at $x=-2$ and $x=0$:
Height at $x=-2$: $y(-2) = 8$
Height at $x=0$: $y(0) = -\frac{1}{4}(0)^2 + 9 = 9$
Area $A_{\text{right}} = \Delta x \cdot (y(-2) + y(0)) = 2 \cdot (8 + 9) = 34$
Step4: Average the two areas
Find the mean of the two areas:
$\text{Average Area} = \frac{A_{\text{left}} + A_{\text{right}}}{2}$
Answer
$\frac{26 + 34}{2} = 30$
Question Image
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Subject
mathematics
Sub Subject
calculus
Education Level
high school
Difficulty
unspecified
Question Type
with image, calculation
Multi Question
No
Question Count
1
Analysis Status
completed
Analyzed At
2026-02-09T19:56:03
OCR Text
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approximate the area of the region bounded by $y = -\\frac{1}{4}x^{2}+9$, the x-axis, $x = -4$, and $x = 0$ by finding the combined area of the rectangles (as shown in each figure) and averaging the results.
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