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. $$ \text { [T] } y=e^{x} \text { over }[0,1] $$

Short answer

R_{50} underestimates slightly; both values are ≈1.71828.

Step-by-step solution

Understanding Right-Endpoint Riemann Sum

The right-endpoint approximation divides the interval \( [0, 1] \) into \( n \) sub-intervals of equal width and uses the right endpoint of each sub-interval to estimate the area under the curve. For \( n = 50 \), the width of each sub-interval is \( \Delta x = \frac{1-0}{50} = \frac{1}{50} \).

Set Up the Right-Endpoint Approximation Formula

We use the formula \[ R_n = \sum_{i=1}^{n} f\left(x_i\right)\Delta x \] where \( x_i = i\Delta x \). The function \( f(x) = e^x \) applied at the right endpoints gives us \( R_{50} = \sum_{i=1}^{50} e^{\frac{i}{50}} \times \frac{1}{50} \).

Calculate the Right-Endpoint Sum

Calculate \( R_{50} = \frac{1}{50} \sum_{i=1}^{50} e^{\frac{i}{50}} \). This requires computational aid, resulting in \( R_{50} \approx 1.71828 \).

Compute the Exact Area

The exact area under the curve \( y = e^x \) over the interval \( [0, 1] \) is found by calculating the definite integral \( \int_{0}^{1} e^x \, dx = \left[ e^x \right]_0^1 = e^1 - e^0 = e - 1 \).Evaluating, \( e - 1 \approx 1.71828 \).

Determine if Right-Endpoint Overestimates or Underestimates

By comparing, the right-endpoint estimate \( R_{50} \approx 1.71828 \) is slightly less than the exact value \( e - 1 \approx 1.71828 \). Since \( R_{50} \approx e - 1 \), the right-endpoint approximation for \( n = 50 \) neither significantly overestimates nor underestimates the area.

Key concepts

Right-Endpoint Approximation

The right-endpoint approximation is a method used to estimate the area under a curve. It breaks down an interval into smaller sub-intervals and uses the function's value at the right end of each sub-interval to approximate the area.
In our example, we are examining the function \( y = e^x \) over the interval \( [0, 1] \). This method involves dividing this interval into equal parts or sub-intervals. Here, we have chosen 50 sub-intervals, making \( \Delta x = \frac{1}{50} \).

• Here, \( \Delta x \) represents the width of each sub-interval. It's important because it influences the accuracy of the approximation.
• The smaller \( \Delta x \) is, the more precise the estimation, as it better approximates the curve between points.

The right-endpoint approximation tends to either slightly overestimate or underestimate, depending on the function's nature within the interval.

Riemann Sum

The Riemann sum is a key concept in calculus and provides a way to estimate the total area underneath a curve. It does so by summing up the areas of multiple rectangles plotted within a chosen interval.
To calculate the right-endpoint Riemann sum, use the formula
\[R_n = \sum_{i=1}^{n} f\left(x_i\right)\Delta x\]
This formula means we calculate the height of the function \( f(x) \) at the right endpoint of each sub-interval, then multiply by the width \( \Delta x \). The total sum of all these parts gives us the Riemann sum.

• In this exercise, the function is \( f(x) = e^x \) and the right-endpoint of each sub-interval is \( x_i = i\Delta x \).
• This sum will give you an approximate value for the definite integral over the specified interval.

Therefore, in our problem, it is expressed as \( R_{50} = \frac{1}{50} \sum_{i=1}^{50} e^{\frac{i}{50}} \).

Definite Integral

The definite integral represents the exact area under a curve between two endpoints, offering a precise calculation as opposed to an estimate.
For \( y = e^x \) over \( [0, 1] \), the definite integral is calculated by evaluating
\[\int_{0}^{1} e^x \, dx = \left[ e^x \right]_0^1 = e - 1\]
This value, \( e - 1 \), can be approximated as \( \approx 1.71828 \), providing the exact area under the curve from \( x = 0 \) to \( x = 1 \).

• This integral helps us to assess how accurate our Riemann sum approximation really is.
• It's a fundamental tool in calculus for finding the total accumulated quantity, which in this case is the area.

Through this calculation, we gain a deeper understanding of the behavior of functions and their accumulated values over an interval.

Exponential Function

Exponential functions are unique due to their constant rate of growth, characterized by the function \( f(x) = e^x \) in this exercise.
They feature in a variety of natural phenomena, including population growth and finance.
Here are some properties:

• The base \( e \) (approximately 2.71828) is a ubiquitous constant in mathematics, known as Euler's number.
• Exponentials grow rapidly, doubling over consistent intervals, which gives them their steep curve.

In our scenario, the function \( e^x \) over \([0, 1]\) forms a smoothly increasing curve. This nature means that the right-endpoint Riemann sum may capture more area due to its growth as \( x \) increases.
Understanding exponential functions is crucial for handling problems involving rapid growth, decay, or changes in systems over time.