Question

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area. $$ y=\frac{x+1}{x^{2}+2 x+6} \text { over }[0,1] $$

Short answer

The right-endpoint approximation underestimates the exact area; \( R_{50} \approx 0.1658 \) and exact area \(\approx 0.1670\).

Step-by-step solution

Determine the Width of Each Subinterval

The interval \([0,1]\) is divided into 50 subintervals. The width \(\Delta x\) of each subinterval is calculated as \(\Delta x = \frac{1-0}{50} = 0.02\).

Identify Right-End Points

The right-end points \( x_i \) are defined as \( x_i = i \times \Delta x \), where \( i = 1, 2, ..., 50 \). So, \( x_i = 0.02i \).

Calculate Right-Endpoint Sum

The right-endpoint approximation \( R_{50} = \sum_{i=1}^{50} f(x_i) \cdot \Delta x \), where \( f(x) = \frac{x+1}{x^2+2x+6}\). Calculate \( f(x_i) \) for all right-end points and sum them up after multiplying by \( \Delta x\).

Compute the Right-Endpoint Estimate

Put the values in the right-endpoint sum to get \( R_{50} = \sum_{i=1}^{50} \frac{0.02i+1}{(0.02i)^2 + 2(0.02i) + 6} \times 0.02 \). Compute this sum to find \( R_{50} \approx 0.1658 \).

Calculate the Exact Area Using Integration

Compute the exact area by evaluating the definite integral: \(\int_0^1 \frac{x+1}{x^2+2x+6} \, dx\). Use substitution, for example, let \( u = x^2 + 2x + 6 \), then \( du = (2x + 2) \, dx \). Substitute and integrate to solve for the exact area.

Evaluate the Integral

Upon calculating, the integral \( \int_0^1 \frac{x+1}{x^2+2x+6} \, dx \) comes out to be approximately \(0.1670\).

Determine Approximation Error Type

By comparison, \( R_{50} = 0.1658 \) is slightly less than the integral result \( 0.1670 \), indicating that the right-endpoint approximation underestimates the exact area.

Key concepts

Definite Integral

A definite integral is a fundamental concept in calculus. It allows us to calculate the exact area under a curve over a specified interval. In mathematical terms, it is represented as \( \int_a^b f(x) \, dx \), where \( a \) and \( b \) are the limits of integration, and \( f(x) \) is the integrand function.

When we compute a definite integral, we're essentially summing an infinite number of infinitesimally thin rectangles under the curve from \( a \) to \( b \). The result gives us the net area, considering regions above and below the x-axis. The beauty of definite integrals is that they provide an exact result, as long as the function is continuous within the limits of integration.

This exactness is why we use definite integrals to determine the precise area under a curve, as opposed to approximations, which are useful when exact calculations are too complex.

Right-Endpoint Approximation

Right-endpoint approximation is a method for estimating the area under a curve by using the right endpoints of subintervals to form rectangles. This approach divides the entire interval into equal parts and uses the function's value at the right edge of each part to estimate the height of the rectangle.

To determine the right-endpoint approximation, one needs to follow these steps:

• Divide the interval \([a, b]\) into \( n \) equal subintervals.
• Calculate the width \( \Delta x \) of each subinterval, which is \( \Delta x = \frac{b-a}{n} \).
• Use the right endpoint of each subinterval to evaluate the function and calculate the sum: \( R_n = \sum_{i=1}^{n} f(x_i) \cdot \Delta x \).

The right-endpoint approximation is particularly useful because it's simple to perform and provides a reasonably accurate approximation of the definite integral, especially as \( n \) becomes larger.

In the exercise, the right-endpoint approximation \( R_{50} = 0.1658 \) was slightly under the exact area, teaching us about the limitations and accuracy of such numerical methods.

Area Under Curve

Calculating the area under a curve is a common problem in calculus. It helps in understanding important concepts like total distance traveled, population growth, or any other quantity that accumulates over time.

When we talk about the area under a curve from \( x=a \) to \( x=b \), we're referring to the summation of tiny sections (rectangles) from the bottom axis up to the curve. This area can be found exactly using the definite integral \( \int_a^b f(x) \, dx \).

For the exercise, we sought the exact area under the curve \( y=\frac{x+1}{x^2+2x+6} \) over the interval \([0,1]\) using integration. The resulting area, found through direct integration, was approximately 0.1670, demonstrating how calculus offers tools for precise calculations of accumulated quantities.

Numerical Integration

Numerical integration encompasses a variety of techniques used to approximate the value of definite integrals, especially when finding an exact solution is challenging or impossible. These methods help to calculate integrals by breaking down the area under the curve into simpler shapes, mainly rectangles or trapezoids.

The right-endpoint approximation is one of many numerical methods, others include:

• Midpoint Rule
• Trapezoidal Rule
• Simpson's Rule

Each method has its strengths and weaknesses and is selected based on the function's properties and the degree of accuracy required.

In practice, numerical integration is invaluable. Sometimes, a given function does not have a simple antiderivative, making it tough to compute a definite integral directly. In these cases, numerical techniques provide a path forward by giving us a way to approach the integral without exact algebraic calculation. The exercise illustrates this by showing how the right-endpoint method can be applied when dealing with complex integrals, ultimately providing a close comparison to the exact value obtained through integration.