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1. 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.

2. 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:
   $\text{Average Area} = \frac{A_{\text{left}} + A_{\text{right}}}{2}$

3. Final answer

   $\frac{26 + 34}{2} = 30$

Réponse

Question Analysis

OCR Text

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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