Problem 10 - 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. $$ y=-x^{2}+5 $$

Step-by-Step Solution

Answer

The estimated area using inscribed rectangles is 8 square units, while the estimated area using circumscribed rectangles is 7 square units.

Step 1: Inscribed Rectangles

First let's deal with the inscribed rectangles. The height of the rectangles should be the function value at the left end of the intervals, which are \([0,1]\) and \([1,2]\). The area is approximated by: \(\sum_{i=0}^{1} f(i) \times 1 = f(0) + f(1)\) where \(f(x) = -x^{2} + 5\). So, it's necessary to substitute 0 and 1 into the equation and sum the results.

Step 2: Circumscribed Rectangles

Now, let's calculate the area using circumscribed rectangles. Here, the height of the rectangles should be the function value at the right end of the intervals, thus, the areas of the rectangles are \(\sum_{i=1}^{2} f(i) \times 1 = f(1) + f(2)\). So, it's necessary to substitute 1 and 2 into the equation and sum the results.

Step 3: Evaluating the Sums

Calculate the areas using the function values found previously. Then, compare the two approximations to understand how the location of height determination (left vs. right ends of the intervals) can affect the Riemann sums and the estimation of the area under the curve.

Key Concepts

Area Under a Curve

Understanding the area under a curve is a foundational concept in calculus. It refers to measuring the space between a curve and the x-axis over a specific interval.
In this exercise, we aim to estimate the area under the curve of the function \(y = -x^2 + 5\) from \(x = 0\) to \(x = 2\).
To approximate this area, we use Riemann sums, which involve dividing the interval into smaller segments, calculating areas using rectangles, and summing the results.
This can be done using inscribed and circumscribed rectangles. Each method provides a different approximation of the area.

Inscribed Rectangles

Inscribed rectangles are formed by using the left endpoint of each subinterval to determine the rectangle's height.
In our problem, with a unit interval from \(0\) to \(2\), we use the points \(x = 0\) and \(x = 1\) to find the height based on the function value at these points. Thus, the height of each rectangle is the function value at \(x = 0\) and \(x = 1\).

• For \(x = 0\), \(f(0) = 5\)
• For \(x = 1\), \(f(1) = -1^2 + 5 = 4\)

Adding these, the estimated area with inscribed rectangles is \(5 + 4 = 9\).
This helps illustrate how using the left endpoints can potentially underestimate the area.

Circumscribed Rectangles

Circumscribed rectangles, on the other hand, use the right endpoint of each subinterval to determine the height.
This changes the base points for our calculations to \(x = 1\) and \(x = 2\).

• For \(x = 1\), \(f(1) = 4\)
• For \(x = 2\), \(f(2) = -2^2 + 5 = 1\)

Therefore, the total estimated area with circumscribed rectangles is \(4 + 1 = 5\).
This typically shows a tendency to overestimate the area when compared to inscribed rectangles.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.
In our exercise, the function \(y = -x^2 + 5\) is a quadratic polynomial, where the highest degree of \(x\) is 2.

• The term \(-x^2\) causes the function to have a downward opening parabola.
• The constant term \(+5\) shifts the entire parabola up by 5 units on the y-axis.

Understanding these functions is crucial to analyze the behavior of the curve, which in turn helps in applying methods like Riemann sums to find the area under the curve over a given interval.