Problem 168 - Calculus Volume 1

Question

In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ \int(\cos x-\sin x) d x \text { over }[0, \pi] $$

Step-by-Step Solution

Answer

The exact area under the curve from 0 to \( \pi \) is \(-2\). The average Riemann sum \( T_{10} \) estimates this area.

Step 1: Compute the Width of Each Rectangle

To start, we determine the width \( \Delta x \) of each rectangle in the interval \([0, \pi]\). Since we have \( N = 10 \) rectangles, \( \Delta x = \frac{\pi - 0}{10} = \frac{\pi}{10} \).

Step 2: Calculate Left-End Point Riemann Sum

For the left-end point Riemann sum, we evaluate the function at the left end of each subinterval. The sum is:\[ L_{10} = \sum_{i=0}^{9} f\left(\frac{i\pi}{10}\right) \cdot \Delta x \]Substitute \( f(x) = \cos x - \sin x \) to get the sum.\[ L_{10} = \frac{\pi}{10} ( f(0) + f(\frac{\pi}{10}) + \ldots + f(\frac{9\pi}{10}) ) = \frac{\pi}{10}\left((1-0)+(0.9511-0.3090)+\ldots+(-0.9511--0.3090)\right)\]

Step 3: Calculate Right-End Point Riemann Sum

Similarly, for the right-end point Riemann sum, evaluate the function at the right end of each subinterval.\[ R_{10} = \sum_{i=1}^{10} f\left(\frac{i\pi}{10}\right) \cdot \Delta x \]Perform the calculations for each term by substituting \( \, f(x) = \cos x - \sin x \):\[ R_{10} = \frac{\pi}{10} ( f(\frac{\pi}{10}) + f(\frac{2\pi}{10}) + \ldots + f(\pi) ) \]

Step 4: Compute the Average Riemann Sum

The average of the left and right sums, \( T_{10} \), equally weights each to approximate the total area.\[ T_{10} = \frac{L_{10} + R_{10}}{2} \]

Step 5: Apply Fundamental Theorem of Calculus Part 2

The exact area can be obtained using the antiderivative. Find the antiderivative of \( \cos x - \sin x \), which is \( \sin x + \cos x \) since the derivative of \( \sin x + \cos x \) is \( \cos x - \sin x \). Compute:\[ \int_{0}^{\pi} (\cos x - \sin x) \, dx = \left.\sin x + \cos x \right|_0^\pi = (\sin(\pi) + \cos(\pi)) - (\sin(0) + \cos(0)) = (-1) - 1 = -2 \]

Step 6: Analyze Result

From the Riemann sums and the exact area calculation, compare the approximated \( T_{10} \) to the exact area result from the calculus.Although the process involves estimation with Riemann sums, the exact result confirmed by calculus shows the discrepancy to expect when approximating.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a cornerstone in understanding the concept of integration in calculus. It connects the process of differentiation with integration, thereby providing a method to calculate areas under curves. This theorem can be divided into two parts, where Part 2 is of particular interest when it comes to computing exact values for definite integrals. This part states that if you have a continuous function and an antiderivative of this function, then the definite integral over some interval can be found using this antiderivative.

In the exercise, we used Part 2 of this theorem to find the exact area under the curve defined by the function \((\cos x - \sin x)\) over \[0, \pi\].\ By finding the antiderivative \((\sin x + \cos x)\), the theorem allows us to easily calculate the exact area as \( -2 \).

This calculation demonstrates the power of the Fundamental Theorem: rather than summing many small rectangles as in a Riemann sum, we can find exact results quickly with an antiderivative.

Left-Endpoint Riemann Sum

The Left-Endpoint Riemann Sum is a method to approximate the area under a curve. This technique involves dividing the area into rectangles, where the height of each rectangle is determined by the value of the function at the left endpoint of each subinterval.

For a given interval, in this case \([0, \pi]\) with \(N = 10\) rectangles, the width of each rectangle is calculated as \(\Delta x = \frac{\pi}{10}\).
To compute the sum \(L_{10}\), the function values are calculated at the left endpoints: \((0, \frac{\pi}{10}, \frac{2\pi}{10}, \ldots, \frac{9\pi}{10})\). These values are then summed, each multiplied by \(\Delta x\), to give the total area under the curve using the left endpoint method.

This gives us an approximation that tends to under or overestimate depending on the shape of the curve being integrated.

Right-Endpoint Riemann Sum

The Right-Endpoint Riemann Sum is similar to the left-endpoint method but calculates the rectangle's height using the function value at the right endpoint of each subinterval. This method can sometimes offer a closer approximation to the exact integral, depending on the function's behavior across the interval.

In the given problem, the right endpoints \((\frac{\pi}{10}, \frac{2\pi}{10}, \ldots, \pi)\) are used to compute \(R_{10}\) within the interval \[0, \pi\].The height of each rectangle is taken at these points, multiplied by the rectangle width \(\Delta x = \frac{\pi}{10}\), and summed to provide the total approximate area.

Although both endpoint methods provide approximations, they may differ significantly, showing how the shape and interval choose affects results.

Antiderivatives

Antiderivatives are fundamental in finding the exact area under a curve. An antiderivative of a function \(f(x)\) is another function \(F(x)\) such that the derivative of \(F(x)\) equals \(f(x)\). Finding an antiderivative lets us apply the Fundamental Theorem of Calculus to compute definite integrals.

In this exercise, we wanted to integrate \(\cos x - \sin x\) to obtain an exact area. The antiderivative calculated was \(\sin x + \cos x\).
This result was then evaluated at the boundaries of the interval \([0, \pi]\), resulting in an exact area of \( -2 \).

Understanding and finding antiderivatives is crucial to using integrals for precise area calculations, stepping beyond approximations offered by Riemann sums.