Problem 41 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{1}^{e^{2}} \int_{\ln x}^{2} f(x, y) d y d x\)

Step-by-Step Solution

Answer

\(\int_{0}^{2} \int_{e^{y}}^{e^{2}} f(x, y) \, dx \, dy \)

Step 1 - Identify the original limits of integration

The given integral is \(\int_{1}^{e^{2}} \int_{\ln x}^{2} f(x, y) \, dy \, dx \).This means that the outer integral variable is \(x\) which ranges from \(1\) to \(e^{2}\), and the inner integral variable is \(y\) which ranges from \(\ln x\) to \(2\).

Step 2 - Sketch the region of integration

To sketch the region of integration:1. Plot the curve \(y = \ln x\).2. Plot the lines \(x = 1\) and \(x = e^{2}\).3. Plot the line \(y = 2\).The region of integration is bounded by \(y = \ln x\), \(y = 2\), \(x = 1\), and \(x = e^{2}\). This region lies above the curve \(y = \ln x\) and below the line \(y = 2\), between \(x = 1\) and \(x = e^{2}\).

Step 3 - Determine the new limits of integration

To reverse the order of integration, the outer integral will now be with respect to \(y\) and the inner integral with respect to \(x\).1. Identify the range of \(y\): from the lowest value (\(\ln 1 = 0\)) to the highest value (\(2\)).2. For a fixed \(y\)-value: \(y = \ln x \Rightarrow x = e^{y}\). So the limits of \(x\) for a fixed \(y\) are from \(x = e^{y}\) to \(x = e^{2}\).

Step 4 - Set up the new integral with reversed order

Using the new limits identified, the integral with the reversed order of integration becomes:\(\int_{0}^{2} \int_{e^{y}}^{e^{2}} f(x, y) \, dx \, dy \).Here, the outer integral is with respect to \(y\) from \(0\) to \(2\), and the inner integral is with respect to \(x\) from \(e^{y}\) to \(e^{2}\).

Key Concepts

Reverse Order of Integration

In integration, switching the order of integration can simplify the process of evaluating a double integral. The given integral is \(\int_{1}^{e^{2}} \int_{\ln x}^{2} f(x, y) \, d y \, d x\). Here, the outer integral is in terms of \(x\), and the inner integral is in terms of \(y\).
By reversing the order of integration, we swap these roles. The new integral will have the outer integral in terms of \(y\) and the inner integral in terms of \(x\). This is helpful when the given limits are more easily expressed in the new order.

Integration Limits

Integration limits define the range over which the integration is performed. In the given problem, the limits for the original integral are:

• \(x\) ranges from 1 to \(e^{2}\)
• \(y\) ranges from \(\ln x\) to 2

These limits describe the bounds of the region we'll integrate over.
When we reverse the order of integration, we must find the new limits for \(y\) and \(x\). First, the outer integral's \(y\)-limits go from \(\ln 1 = 0\) to 2. For a fixed \(y\), \(x\) ranges from \(e^{y}\) to \(e^{2}\).

Region of Integration

The region of integration is the area defined by the given limits. To understand this:

• Sketch the curve \(y = \ln x\)
• Draw the vertical lines \(x = 1\) and \(x = e^{2}\)
• Draw the horizontal line \(y = 2\)

The region lies above the curve \(\ln x\) and below the line \(y = 2\), between \(x = 1\) and \(x = e^{2}\).
Visualizing this region helps in understanding the bounds for switching the order of integration.

Calculus Steps

Breaking down the calculus steps to reverse the order involves:

• Identifying original limits: Outer \(x\) from 1 to \(e^{2}\), Inner \(y\) from \(\ln x\) to 2
• Sketching the region of integration to visualize
• Setting new limits: Outer \(y\) from 0 to 2, Inner \(x\) from \(e^{y}\) to \(e^{2}\)
• Writing the new integral: \(\int_{0}^{2}\int_{e^{y}}^{e^{2}} f(x, y) \, dx \, dy\)

Using these steps ensures the integral's value remains unchanged, providing better insight or simplifying computation.