Problem 47 - Calculus Volume 1

Question

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. $$ [T]y=\cos x, y=e^{x}, x=-\pi, \text { and } x=0 $$

Step-by-Step Solution

Answer

The exact area cannot be determined analytically, approximately 3.203.

Step 1: Identify the Functions and Limits

We have two functions, \(y = \cos x\) and \(y = e^x\), and we need to find their intersection points within the interval from \(x = -\pi\) to \(x = 0\). These intersection points will be used as bounds for integration, along with \(x = -\pi\) and \(x = 0\).

Step 2: Find the Intersection Points

To find the intersection points of \(y = \cos x\) and \(y = e^x\), we set them equal: \[\cos x = e^x\] This equation does not have a simple analytical solution, so we use a calculator to approximate the intersection points in the interval \(x = -\pi\) to \(x = 0\). The approximate point of intersection is \(x \approx -0.739\).

Step 3: Determine the Higher Function on the Interval

For each subinterval defined by \([-\pi, -0.739]\) and \([-0.739, 0]\), determine which function is above the other. - On the interval \([-\pi, -0.739]\), \(\cos x\) is above \(e^x\). - On the interval \([-0.739, 0]\), \(e^x\) is above \(\cos x\).

Step 4: Integrate to Find the Area

We integrate the difference between the top and bottom functions over each subinterval. 1. For \(x \in [-\pi, -0.739]\): \[\int_{-\pi}^{-0.739} (\cos x - e^x) \, dx\] 2. For \(x \in [-0.739, 0]\): \[\int_{-0.739}^{0} (e^x - \cos x) \, dx\]

Step 5: Calculate the Integrals

Compute the integrals from Step 4: 1. \[\int (\cos x) \, dx = \sin x\] and \[\int (e^x) \, dx = e^x\]. Use these to calculate: - \[\left[ \sin x - e^x \right]_{-\pi}^{-0.739}\] - \[\left[ e^x - \sin x \right]_{-0.739}^{0}\].

Step 6: Compute the Total Area

Evaluate the definite integrals from Step 5: - For \([-\pi, -0.739]\): Calculate \((\sin(-0.739) - e^{-0.739}) - (\sin(-\pi) - e^{-\pi})\). - For \([-0.739, 0]\): Calculate \((e^0 - \sin 0) - (e^{-0.739} - \sin(-0.739))\). Add the absolute values of these areas to find the total area of the region.

Key Concepts

Intersection Points

In solving problems with areas between curves, finding the intersection points of the functions is crucial. These points indicate where the curves meet and help define the boundaries for integration. For the given functions, \( y = \cos x \) and \( y = e^x \), finding where they intersect involves equating the two equations: \( \cos x = e^x \).

This equation does not have a straightforward algebraic solution, making it necessary to use numerical methods, such as a calculator, to approximate the intersection points. In our case, this leads us to find an intersection point around \( x \approx -0.739 \) within the interval \( x = -\pi \) to \( x = 0 \).

These intersection points divide the interval into subintervals, each having specific bounds for integration, highly important for calculating the exact area between the curves.

Integration

Integration is essential in determining the area between curves. Once the interval is divided based on the intersection points, each segment can be evaluated separately by integrating over it. The overall strategy involves:

• Identifying which function is on top within each subinterval.
• Subtracting the lower function from the upper function within the integral.

Mathematically, this becomes applying definite integrals over each subinterval. For example, if one segment requires evaluating \( \int_{a}^{b} (\text{upper function} - \text{lower function}) \, dx \), it's important that the difference accounts for which function is actually higher in that specific interval.

Integrals are calculated using known antiderivatives like \( \int \cos x \, dx = \sin x \) and \( \int e^x \, dx = e^x \), which further simplifies evaluating the definite integrals over each segment.

Trigonometric Functions

Trigonometric functions, like \( y = \cos x \), are recurrent in mathematical problems involving curves. The cosine function is periodic with a typical wavy pattern, undulating between -1 and 1. Understanding the graphical behavior of these functions is crucial when determining which function lies above or below another within specified intervals.

In the interval \( [ -\pi, 0 ] \), \( y = \cos x \) has a range of behaviors, affecting the outcome of the area calculation. Between the intersection point and \( x = -\pi \), \( \cos x \) remains above \( e^x \).

Familiarity with trigonometric identities and the unit circle can assist in interpreting results and confirming that calculations align with expected behavior.

Exponential Function

The function \( y = e^x \) is an exponential function known for its rapid growth rate. Unlike trigonometric functions, exponential functions don't oscillate; they increase consistently. This characteristic becomes apparent when analyzing which function is higher.

Between the intersection point \( x = -0.739 \) and \( x = 0 \), \( e^x \) surpasses \( \cos x \). Its growth is exponential, meaning that as \( x \) becomes less negative, \( e^x \) quickly becomes the dominant function above the trigonometric \( \cos x \).

When computing areas, understanding this rapid increase helps predict regions where \( e^x \) dictates the top boundary, influencing the formulation of integrals and confirming the correct calculation of areas.