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

Short answer

R50 overestimates the area compared to the exact integral, which is $1-\frac{1}{e}$.

Step-by-step solution

Understanding Right-Endpoint Approximations

A right-endpoint approximation uses the function value at the right endpoint of each subinterval to approximate the area under the curve. In this case, our function is $y=e^{-x}$ over the interval $[0,1]$.

Determine the Subinterval Width

Divide the interval $[0,1]$ into 50 equal subintervals. The width $\mathrm{\Delta }x$ of each subinterval is given by $\mathrm{\Delta }x=\frac{b-a}{n}=\frac{1-0}{50}=0.02$.

Calculate Right-End Values

For each subinterval, the right endpoint is $x_i=i\cdot \mathrm{\Delta }x$ for $i=1,2,...,50$. Evaluate the function at each right endpoint: $e^{-x_i}$.

Sum the Rectangles' Areas

The right endpoint approximation for the area, $R_{50}$, is calculated by summing the areas of the rectangles: $R_{50}=\sum _{i=1}^{50}{e}^{-x_i}\cdot \mathrm{\Delta }x$. All calculations involve arithmetical simplification before adding each product.

Calculate the Exact Area with the Integral

The exact area under the curve from 0 to 1 is given by evaluating the definite integral ${\int }_0^1{e}^{-x}dx$. This integral evaluates to $-e^{-x}{|}_0^1$, which simplifies to $-e^{-1}+e^0=1-\frac{1}{e}$.

Determine Overestimation or Underestimation

Since $e^{-x}$ is a decreasing function over $[0,1]$, using the right endpoint values overestimate the area under the curve compared to the exact integral value.

Key concepts

Right-Endpoint Approximation

The right-endpoint approximation is a method used in calculus to estimate the area under a curve on a given interval. By dividing the interval into equal subintervals, this method uses the value of the function at the right endpoint of each subinterval to form rectangles that approximate the area. This method is straightforward and forms the foundation for understanding numerical integration techniques.
For example, if you are trying to approximate the area under the curve for a function on the interval [0, 1], you divide the interval into smaller intervals, like fifty subintervals, and use the function value at the right edge of each to form a rectangle. You then sum up the areas of these rectangles.
This approximation provides an estimate of the area, which might differ from the exact area. A closer look at this difference helps in understanding the accuracy and efficiency of this method.

Definite Integral

The concept of the definite integral is at the heart of calculus and is used to find the exact area under a curve between two points on the x-axis. For a function like $y=e^{-x}$ over the interval [0, 1], we evaluate the definite integral to calculate the precise area.
The process involves finding the antiderivative of the function and then applying the fundamental theorem of calculus to solve it over the specific limits. In this exercise, the definite integral is ${\int }_0^1{e}^{-x}dx$, which evaluates to $-e^{-x}{|}_0^1$ leading to a solution of $1-\frac{1}{e}$.
This calculational method eliminates the need for approximation, giving an exact measure of the area under the curve, and thus acts as a benchmark to compare with numerical approximations.

Overestimation and Underestimation

When using numerical methods like the right-endpoint approximation, there is always a possibility of overestimating or underestimating the true area under the curve. This is due to the nature of the function being approximated.
In the case of the function $e^{-x}$, which decreases over the interval [0, 1], using right endpoints creates rectangles that tend to overshoot the curve. Thus, the approximation yields an overestimated area.
Recognizing when a function leads to overestimation or underestimation is key in improving the accuracy of numerical approaches. It allows for adjustments and better selections of approximation techniques, tailored to provide a close measure to the true integral.

Exponential Function

The exponential function $e^{-x}$ plays a significant role in both mathematics and applied sciences. In calculus, it is a common subject for problems focused on integration and differentiation due to its unique properties.
Characteristics of the exponential function, particularly $e^{-x}$, makes it interesting for analysis as it is a decreasing, continuous function on the interval [0, 1]. This impacts the behavior of approximation methods like the right-endpoint method, leading typically to overestimated results.
Understanding its behavior, like how it decreases rapidly as $x$ increases, is crucial for accurately tackling problems involving exponential functions in calculus.