Problem 370 - Calculus Volume 1

Question

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate \(R_{50}\) and solve for the exact area. $$y=2^{x} \text { over }[-1,0]$$

Step-by-Step Solution

Answer

The right-endpoint approximation \( R_{50} = 0.3493 \) underestimates the exact area \( 0.72135 \).

Step 1: Understand the Problem

We need to calculate the right-endpoint approximation, denoted as \( R_{50} \), for the function \( y=2^x \) over the interval \([-1, 0]\). Then, compare it with the exact area under the curve on the same interval to determine if the approximation overestimates or underestimates the true area.

Step 2: Calculate the Exact Area

First, calculate the exact area under the curve \( y=2^x \) from \( x=-1 \) to \( x=0 \) using the definite integral:\[\int_{-1}^{0} 2^x \, dx\]The integral of \( 2^x \) is \( \frac{2^x}{\ln(2)} \). Evaluate from \( -1 \) to \( 0 \):\[\left[ \frac{2^x}{\ln(2)} \right]_{-1}^{0} = \frac{2^0}{\ln(2)} - \frac{2^{-1}}{\ln(2)} = \frac{1}{\ln(2)} - \frac{0.5}{\ln(2)} = \frac{0.5}{\ln(2)}\]

Step 3: Set Up the Right-Endpoint Approximation

For the right-endpoint approximation \( R_{50} \), divide the interval \([-1,0]\) into 50 equal subintervals each of width \( \Delta x = \frac{1}{50} = 0.02 \). The right endpoints are: \( -0.98, -0.96, ..., 0 \). The function values at these points are used to approximate the area.

Step 4: Calculate \( R_{50} \)

Evaluate \( R_{50} \) using the formula:\[R_{50} = \sum_{i=1}^{50} f(x_i^*) \Delta x\]where \( x_i^* = -1 + i \times 0.02 \) and \( f(x) = 2^x \). Calculating gives:\[R_{50} \approx 0.3493\]

Step 5: Compare \( R_{50} \) with the Exact Area

The exact area calculated is \( \frac{0.5}{\ln(2)} \approx 0.72135 \). Since \( R_{50} = 0.3493 \), which is less than the exact area, the right-endpoint approximation underestimates the exact area under \( y=2^x \) on \([-1,0]\).

Key Concepts

Definite Integral

In calculus, a definite integral is a fundamental concept used to calculate the total area under a curve within a certain interval on the x-axis. It's symbolized by an integral sign with limits, representing the start and end of the interval. The process involves not only finding how much space is between the curve and the x-axis but also capturing the net area, considering parts above and below the axis separately.
When calculating a definite integral, the function's antiderivative is determined first. By applying the limits of integration, we evaluate the function at the upper limit and subtract the value of the function at the lower limit. For example, the definite integral of an exponential function like \( y = 2^x \) from \( x = -1 \) to \( x = 0 \) helps us find the precise area under this exponential curve over that specific interval.

Right-Endpoint Approximation

The right-endpoint approximation is a numerical method to estimate the area under a curve when calculating the exact area might be complex or infeasible. It involves dividing the interval into a certain number of smaller subintervals; in this case, it's 50 subintervals.
For each subinterval, we calculate the function value at the right endpoint and multiply it by the width of the subintervals. This width, denoted as \( \Delta x \), is uniform and found by dividing the total interval length by the number of subintervals.
Then, by adding up the areas of all these thin rectangular segments, represented by these function values, we get the total approximate area under the curve. In our exercise, a right-endpoint approximation with 50 subintervals yields an estimate of the area under the curve which tends to underestimate as curves usually rise, and thus the rectangles based on right endpoints start from below the curve.

Exponential Functions

Exponential functions, like \( y = 2^x \), are vital in mathematics because they model situations where growth or decay accelerates rapidly. These functions have the form \( a^x \), where \( a \) is a constant base, greater than zero. Exponential functions typically show rapid change, which makes analyzing their behavior crucial.
When working with exponential functions, calculating the integral involves the exponential rule. The integral of \( a^x \) is \( \frac{a^x}{\ln(a)} \), where \( \ln(a) \) is the natural logarithm of the base \( a \). Evaluating an integral involving exponential functions helps us understand the area dynamics over an interval, further allowing us to compare it with approximation methods like the right-endpoint to see the accuracy by which they estimate the area.

Area under a Curve

The area under a curve provides significant information about the behavior and trend of a function within a specified interval. This area is closely related to integral calculus, as finding it requires solving a definite integral.
For functions like \( y = 2^x \), knowing the area under the curve from \( x = -1 \) to \( x = 0 \) allows us to grasp how much the exponential function grows or lessens over this interval. This essential concept not only solves mathematical problems but also finds real-world applications in physics, engineering, and economics where understanding the total quantity between given values is crucial.
Thus, mastering how to compute the area under a curve, whether through exact methods using integrals or approximate numerical methods like right-endpoint approximation, is a cornerstone in developing strong calculus skills.