Problem 15 - Algebra 2

Question

Write and evaluate a sum to estimate the area under each curve for the domain \(0 \leq x \leq 2\) . a. Use inscribed rectangles 1 unit wide. b. Use eircumscribed rectangles 1 unit wide. $$ y=4-\frac{1}{4} x^{2} $$

Step-by-Step Solution

Answer

The total estimated area under the curve defined by the equation \(y=4-\frac{1}{4} x^{2}\) in the domain \(0 \leq x \leq 2\) using inscribed rectangles is the sum of the areas of the rectangles for the given domain using the height at the left end of each rectangle. Similarly, the total estimated area using circumscribed rectangles is the sum of areas using the height at the right end of the rectangle.

Step 1: Define the Shape

This is a downward Parabola with domain \(0 \leq x \leq 2\). Therefore, inscribed rectangles are on top of the curve while circumscribed rectangles lie underneath.

Step 2: Inscribed Rectangles

For the inscribed rectangles, the height of the rectangle is equal to the function value at the left-end point of the rectangle: \[h_{i}=f(x_{left})=4-\frac{1}{4} x_{left}^{2}\]. Since each rectangle has width of 1, the area is \[A_{i} = h_{i} \times 1\]. Then, summing areas of all rectangles from x=0 to 2 \[A_{inscribed} = \sum_{x=0}^{2} A_{i}\]

Step 3: Circumscribed Rectangles

For the circumscribed rectangles, the height is the function value at the right-end point of the rectangle: \[h_{c}=f(x_{right})=4-\frac{1}{4} x_{right}^{2}\]. The area for each rectangle is \[A_{c} = h_{c} \times 1\]. Then, summing areas of all rectangles from x=0 to 2 \[A_{circumscribed} = \sum_{x=1}^{2} A_{c}\]

Step 4: Calculate and Sum

Calculate the areas for inscribed and circumscribed rectangles and add the results

Key Concepts

Inscribed Rectangles

In calculus, inscribed rectangles are used to approximate the area under a curve by fitting rectangles under the curve. For a downward parabola like the one defined by the equation \(y=4-\frac{1}{4}x^{2}\), using rectangles of equal width helps in simplifying our calculations. The width here is 1 unit.

To find the area of each inscribed rectangle, we calculate the height at the left endpoint of the interval:

• Use the left value of \(x\) of each interval: \(0\) and \(1\).
• Plug it into the function \(f(x) = 4-\frac{1}{4}x^{2}\) to find each height.
• The area of each rectangle is then \(h_{i} \times 1\), since the width is 1.
• Sum up all these areas to find the total approximated area under the curve.

Circumscribed Rectangles

Circumscribed rectangles, on the other hand, fit on top of the curve. When calculating the area using circumscribed rectangles, the height is determined at the right endpoint of each interval. This means using the next point's \(x\) value, making the rectangles slightly larger as they extend over the curve.

Here's how this is done:

• Use the right value of \(x\) for each interval: \(1\) and \(2\).
• Plug these into \(f(x) = 4-\frac{1}{4}x^{2}\) to get the height of each rectangle.
• Again, each rectangle has a width of 1, so the area is calculated as \(h_{c} \times 1\).
• Add up these areas for the total approximation.

Parabolas

Parabolas are a type of curve described by a quadratic equation. In our example, the equation \(y=4-\frac{1}{4}x^{2}\) represents a downward-facing parabola. This shape is symmetric around the vertical axis and opens downwards due to the negative sign before the \(x^{2}\) term.

Key features of parabolas include:

• The vertex, which in this function is at the highest point \((x=0, y=4)\).
• Its symmetry, which makes calculations predictable as the curve behaves consistently on either side of the vertex.
• The role of the quadratic term coefficient (here \(-\frac{1}{4}\)) which affects the width of the parabola. A smaller absolute value here means the curve is wider.

Riemann Sums

Riemann sums are a crucial concept in understanding how to approximate the area under a curve, which is a fundamental step in integral calculus. By dividing the area into shapes like inscribed and circumscribed rectangles, we can form sums that approximate the area more accurately.

Process of using Riemann sums:

• Divide the curve into intervals, choosing points on these intervals to calculate rectangle heights.
• Calculate areas of rectangles using formulas derived from these points, as seen with the function \(f(x)\).
• Sum the areas of all these rectangles to find an approximate area under the curve.
• The more intervals and rectangles we use, the closer our approximation gets to the true area.