Problem 11 - 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. $$ g(x)=x^{2}+1 $$

Step-by-Step Solution

Answer

The estimated area under the curve \(g(x) = x^{2}+1\) from \(x=0\) to \(x=2\) using inscribed rectangles is 3. Using circumscribed rectangles, the estimated area is 7.

Step 1: Formularise the Problem

First, identify the function, here it's \(g(x)=x^{2}+1\). The domain is \(x \in [0, 2]\). The width of each rectangle will be 1. Two methods will be used to estimate the area under the function, using inscribed rectangles and using circumscribed rectangles.

Step 2: Inscribed Rectangles

Inscribed rectangles mean that the rectangle for each subinterval lies entirely below the curve. The height of each rectangle is given by the function value at the left end of the subinterval. Here, there are 2 rectangles. The first rectangle's height is \(g(0)=0^{2}+1=1\) and the height of the second rectangle is \(g(1)=1^{2}+1=2\). The area is thus estimated as \(\sum_{i=1}^{2} Area_{i} = \sum_{i=1}^{2} Width_{i}*Height_{i} = 1*1 + 2*1 = 3\).

Step 3: Circumscribed Rectangles

Circumscribed rectangles mean that the rectangle for each subinterval lies entirely above the curve. The height of each rectangle is given by the function value at the right end of the subinterval. The first rectangle's height is \(g(1)=2\), and the second rectangle's height is \(g(2)=4+1=5\). The area is thus estimated as \(\sum_{i=1}^{2} Area_{i} = \sum_{i=1}^{2} Width_{i}*Height_{i} = 2*1 + 5*1 = 7\).

Key Concepts

Inscribed Rectangles

To estimate the area under a curve, inscribed rectangles can be used, which are rectangles placed inside the curve. Here’s how it works:

• An inscribed rectangle touches the curve from below. This rectangle will be smaller than the actual area under the curve, providing an underestimation.
• For each segment defined in your interval, you calculate the height of the rectangle using the function value at the left endpoint of that segment.
• The width of each rectangle is given, so the area can be calculated by multiplying the width by the height obtained.

In our example, the function used was \(g(x) = x^2 + 1\), and each rectangle had a width of 1.Starting from \(x=0\), the height of the first rectangle is \(g(0) = 1\), and for \(x=1\), the second rectangle height is \(g(1) = 2\). Thus, the total estimated area using inscribed rectangles is 3 because: \[ Area = 1 \times 1 + 1 \times 2 = 3.\]

Circumscribed Rectangles

Circumscribed rectangles work slightly differently. These rectangles instead aim to entirely cover the area under the curve. Here’s what sets them apart:

• The rectangle surpasses the curve, typically touching the curve along the top; this means the calculated area is an overestimate.
• To find the rectangle's height, utilize the function value at the right endpoint of each segment.
• This approach ensures the entire curve segment is covered, providing a boundary from above.

With the example function \(g(x) = x^2 + 1\), and given rectangle width of 1, we start at \(x=0\) up to \(x=2\). For the first rectangle, height is \(g(1) = 2\), and for the second, \(g(2) = 5\). This results in an estimated total area of:\[ Area = 1 \times 2 + 1 \times 5 = 7.\]This approach provides an estimation that overshoots the actual area under the curve.

Area Under A Curve

The concept of finding the area under a curve is foundational in calculus and involves summing up small pieces that fill the space between the curve and the x-axis. Here are some key points:

• The area under a curve can be approached by various summation methods; Riemann sums are among the most common.
• In practice, this area represents quantities such as distance or accumulated quantity depending on the problem context.
• While different methods, such as using inscribed or circumscribed rectangles, may highlight different aspects of the curve, they lay the groundwork for integral calculus.

In our exercise, both inscribed and circumscribed rectangles served as finite approximations of the space beneath \(g(x) = x^2 + 1\). As we increase the number of smaller rectangles (thus narrowing their width), we get closer to the exact integral value of the region. Hence, understanding these rectangle strategies paves a mighty strong path towards grasping integral calculus concepts. Through this, learners grasp how calculus transitions from simple geometrical thinking to profound mathematical insights.