Problem 12 - 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. $$ y=-x^{2}+4 $$

Step-by-Step Solution

Answer

The estimate of the area under the curve using inscribed rectangles is 7 square units, and using circumscribed rectangles is 11 square units.

Step 1: Find the values of y

For each integer value of \(x\) in the range 0 to 2, we calculate the corresponding value of y in the function \(y=-x^{2}+4\). These are (0,4), (1,3), and (2,0).

Step 2: Compute area using inscribed rectangles

In inscribed rectangles, we use the function's minimum value in the given interval as height. Here, the width is 1, and the heights are the y-values. Then the area A (estimate) would be: \[A = 1*(4 + 3) = 7\]

Step 3: Compute area using circumscribed rectangles

In circumscribed rectangles, the function's maximum value in the given interval is used as height. Here, the widths are 1, and the heights are the y-values. Then the area A (estimate) would be: \[A = 1*(4 + 4 + 3) = 11\]

Key Concepts

Inscribed Rectangles

When estimating the area under a curve using inscribed rectangles, each rectangle is drawn within the curve. This means the top of the rectangle touches the curve at the lowest point within the given interval.
In our example, we are finding the area under the polynomial function given by \(y = -x^{2} + 4\), from \(x = 0\) to \(x = 2\).

The process involves:

• Finding the smallest \(y\)-value within each subinterval to use as the rectangle's height.
• For intervals \([0,1)\) and \([1,2)\) in our example, these heights are 4 (from \(y(0)\)=4) and 3 (from \(y(1)\)=3) respectively because 3 is less than 0 at \(y(2)\).
• Multiplying each height by the width of the rectangles (1 unit in our case) and summing these areas gives the total estimated area.

Thus, the area under the curve using inscribed rectangles is estimated to be \(A = 1 \times (4 + 3) = 7\) square units.

Circumscribed Rectangles

Circumscribed rectangles estimate area by extending beyond or surrounding the curve. The top of the rectangle reaches the highest point of the curve within the interval. This provides an overestimate of the true area.

Let's consider the polynomial function \(y = -x^{2} + 4\) from \(x = 0\) to \(x = 2\).
To use circumscribed rectangles effectively:

• Identify the highest \(y\)-value at each subinterval: For \([0,1)\) the height would be 4, for \([1,2)\) the height is 4 as well since it doesn’t drop below 3 in 0 to 1, and not yet hit 0.
• Multiply each height by the width (1 unit) and sum: This yields
• \(A = 1 \times (4 + 4 + 3) = 11\), which includes the extra block above \([1,2)\).

Using these rectangles, the total overestimated area under the curve is 11 square units.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole-number powers, combined using arithmetic operations. They are quite flexible and can represent a wide variety of curves.
In the context of area estimation, polynomial functions often describe the curves we're interested in.

The function \(y = -x^{2} + 4\) is a simple quadratic, meaning its highest degree is \(x^2\).
Here are key characteristics of this type:

• The graph is a parabola, opening downwards, due to the negative coefficient of \(x^2\).
• The vertex is the highest point, at \((0,4)\) in this case, because
• the \(x\)-term is zeroed out and the constant provides the maximum height.
• This curve intersects the \(x\)-axis at roots or points \((2,0)\) and \((2,-2)\), where the value of \(y\) becomes zero.

Understanding these components is essential for correctly estimating areas and interpreting intersections with the axes.