Problem 10 - University Calculus: Early Transcendentals

Question

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The lines \(y=1-x\) and \(y=2\) and the curve \(y=e^{x}\).

Step-by-Step Solution

Answer

The area of the region is approximately 0.252 square units.

Step 1: Identify Intersection Points

First, find the points of intersection between the lines and the curve. Set the equations equal and solve for \( x \). Solve \( 1-x = e^x \) and \( 2 = e^x \). The intersection with \( y=1-x \) requires numerical methods or plotting to solve, approximately \( x \approx 0.567 \). The intersection \( 2 = e^x \) gives \( x = \ln(2) \approx 0.693 \).

Step 2: Sketch the Region

Plot the lines \( y = 1-x \), \( y = 2 \), and the curve \( y = e^x \). Identify the region bounded by these plots. The region of interest is on the interval \( [0.567, 0.693] \) on the x-axis, above the curve \( y=e^x \) and below the line \( y=2 \).

Step 3: Set Up the Double Integral

Set up the double integral to find the area of the region. Since the region is bounded above by \( y = 2 \) and below by \( y = e^x \), and spans from \( x = 0.567 \) to \( x = 0.693 \), the integral becomes: \[ \int_{0.567}^{0.693} \int_{e^x}^{2} dy \, dx \].

Step 4: Evaluate the Inner Integral

Evaluate the inner integral \( \int_{e^x}^{2} 1 \, dy \). This simplifies to \( [y]_{e^x}^{2} = 2 - e^x \).

Step 5: Evaluate the Outer Integral

Substitute the result from the inner integral into the outer integral: \[ \int_{0.567}^{0.693} (2 - e^x) \, dx \]. Compute this by evaluating: \[ \int (2) \, dx - \int (e^x) \, dx \]. Which becomes: \[ (2x - e^x) \bigg|_{0.567}^{0.693} \].

Step 6: Compute the Definite Integral

Substitute the bounds into \( 2x - e^x \): Calculate: \( (2 \times 0.693 - e^{0.693}) - (2 \times 0.567 - e^{0.567}) \). This results in approximately \( 0.252 \).

Key Concepts

Intersection Points

To understand the interplay between the given equations and curves, we first solve for their intersection points. Finding these points involves solving equations like \(1-x = e^x\) and \(2 = e^x\). These intersection points help us identify the limits of the region. For the equation \(1-x = e^x\), a numerical method can approximate that \(x \approx 0.567\). Similarly, solving \(2 = e^x\) yields \(x = \ln(2) \approx 0.693\).

• Intersection between \(1-x\) and \(e^x\) is approx \(x \approx 0.567\).
• Intersection between \(y=2\) and \(e^x\), \(x = \ln(2) \approx 0.693\).

These intersection points define the boundaries of our region over the x-axis.

Bounded Region

Once intersection points are calculated, sketching helps visualize the bounded region. The area lies on the x-interval [0.567, 0.693], above the curve \(y = e^x\) and below the line \(y = 2\). This visualization ensures that we correctly set up the limits for integration.To effectively sketch:

• Draw lines \(y = 1-x\) and \(y = 2\).
• Draw the curve \(y = e^x\).

Where they intersect marks the region. This bounded region is crucial to accurately determine the area using integrals.

Iterated Integral

Setting up an iterated integral allows us to compute the area of the bounded region. The iterated integral breaks down the calculation into manageable parts by evaluating the area in slices.The process involves determining and setting the right limits:

• Vertical limits: between the curve \(y = e^x\) (lower) and the line \(y = 2\) (upper).
• Horizontal limits: from \(x=0.567\) to \(x=0.693\).

The double integral expression becomes:\[\int_{0.567}^{0.693} \int_{e^x}^{2} dy \, dx\]This setup methodically confines the area under examination, which simplifies subsequent evaluation steps.

Numerical Methods

Numerical methods come into play when analytic solutions are complex or impractical. While \(x = \ln(2)\) was straightforward from \(2 = e^x\), the equation \(1-x = e^x\) required more effort.Approaches like numerical approximation or graphing can help:

• Use iterative methods like Newton's or graphical solutions.
• Software tools can provide quick, precise approximations.

These methods are valuable in obtaining solutions where algebra alone might not easily suffice. Incorporating numerical strategies expands our toolbox for complex function analysis.