Problem 14 - 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. $$ h(x)=5 x^{2} $$

Step-by-Step Solution

Answer

The estimated area under the curve using inscribed rectangles is 25 square units while the estimated area using circumscribed rectangles is 70 square units.

Step 1: Computing for Inscribed Rectangles

Using inscribed rectangles involves taking the left endpoint of each subinterval as the x-value for the height of the rectangle. With three rectangles each 1 unit wide, evaluate the function at 0, 1, and 2. That yields \(h(0) = 5*0^{2} = 0\), \(h(1) = 5*1^{2} = 5\), and \(h(2) = 5*2^{2} = 20\). The area under the curve using inscribed rectangles is given by \(\sum_{i=0}^{n}f(x_{i-1})\Delta x\). In this case, \(\Delta x = 1\) since the rectangles are 1 unit wide. So, overall, using inscribed rectangles, the area is \(1*0 + 1*5 + 1*20 = 25\) square units.

Step 2: Computing for Circumscribed Rectangles

Using circumscribed rectangles involves taking the right endpoint of each subinterval as the x-value for the height of the rectangle. Evaluate the function at 1, 2, and 3 respectively. That yields \(h(1) = 5*1^{2} = 5\), \(h(2) = 5*2^{2} = 20\), and \(h(3) = 5*3^{2} = 45\). The area under the curve using circumscribed rectangles is given by \(\sum_{i=0}^{n}f(x_{i})\Delta x\). Just as before, \(\Delta x = 1\) since the rectangles are 1 unit wide. So, overall, the area under the curve using circumscribed rectangles is \(1*5 + 1*20 + 1*45 = 70\) square units.

Key Concepts

Inscribed Rectangles

Inscribed rectangles are a method used to approximate the area under a curve by using rectangles whose heights are determined by the left endpoints of subintervals of the domain. This method tends to underestimate the area because the rectangles fit below the curve, especially when the function is increasing within the interval. To estimate the area under the curve with inscribed rectangles:

• Divide the domain into equal widths. In this example, each rectangle is 1 unit wide.
• Determine the left endpoint for each subinterval, which will serve as the height of each rectangle.
• Calculate the function value at each left endpoint to find the height of each respective rectangle.
• Sum up the areas of these rectangles, calculated by multiplying the width of a rectangle by its height.

For instance, if using three inscribed rectangles over the domain from 0 to 2, evaluate the function at the points 0, 1, and 2. Then calculate the area by summing up each rectangle's area: \[ 1\times h(0) + 1\times h(1) + 1\times h(2) = 0 + 5 + 20 = 25 \]square units. Here, each function result gives you the respective rectangle heights.

Circumscribed Rectangles

Circumscribed rectangles provide another classic approach for approximating areas under curves. Unlike inscribed rectangles, circumscribed rectangles overestimate the area because they extend above the curve. This method uses the right endpoints of each subinterval to determine the rectangle heights. Here's how you can evaluate using circumscribed rectangles:

• Divide the domain into equal widths, ensuring consistency similar to inscribed rectangles.
• Select the right endpoint for each subinterval to determine the rectangle's height.
• Calculate the function value at each right endpoint, which gives you the height of each rectangle.
• Add up all the rectangle areas found using width times height.

If you are calculating the area with circumscribed rectangles for a polynomial function over the domain from 0 to 2:\[1\times h(1) + 1\times h(2) + 1\times h(3) = 5 + 20 + 45 = 70 \] square units. This sum indicates the total area when evaluating at points 1, 2, and 3.

Polynomial Functions

Polynomial functions, like the one given as \(h(x)=5x^2\), are mathematical expressions involving variable powers summed together, typically with coefficients. These functions can take various forms, such as linear, quadratic, cubic, and higher degrees. Here's a basic primer:
- A polynomial of degree \(n\) generally looks like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\).- The degree of a polynomial is the highest power of the variable \(x\) present.
For the given problem, \(h(x)=5x^2\) is a quadratic polynomial (degree 2), with a parabolic shape opening upwards. The curve's nature affects how accurately various Riemann sum approaches approximate areas.
- Quadratic polynomials often lead to more "curved" area approximations.- As \(x\) increases, the function \(5x^2\) grows rapidly due to the squared term, leading to steeper slopes and greater areas under the curve.
Remember, these functions can model several natural phenomena, making their study critical for calculus applications, such as in physics and engineering.