Problem 45 - University Calculus: Early Transcendentals

Question

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{3} \int_{1}^{e^{y}}(x+y) d x d y$$

Step-by-Step Solution

Answer

The reversed integral is \( \int_{1}^{e^3} \int_{0}^{\ln{x}} (x+y) \, dy \, dx \).

Step 1: Understanding the Region of Integration

The given integral is \( \int_{0}^{3} \int_{1}^{e^{y}}(x+y) \, dx \, dy \). This signifies an area in the xy-plane. For each y in [0, 3], x ranges from 1 to \( e^y \). The region is bounded by the lines y=0, y=3, x=1, and the curve \( x = e^y \).

Step 2: Sketching the Region

To sketch the region, plot the straight vertical line x=1 and the curve \( x = e^y \). The region is bounded vertically by y = 0 and y = 3. The intersection of x=1 and the curve happens when \( 1 = e^y \), solving this gives y=0. So the region is bounded between x=1, \( x = e^y \), and y from 0 to 3.

Step 3: Finding the New Ranges for Reversing Integration

For the new order of integration, we need to integrate with respect to y first. Observe that for x in [1, e^3], y ranges from 0 to \( \ln{x} \), given that the curve is \( x = e^y \), thus \( y = \ln{x} \).

Step 4: Writing the Reversed Double Integral

The integral with reversed order of integration becomes \( \int_{1}^{e^3} \int_{0}^{\ln{x}} (x+y) \, dy \, dx \). Now, integration would be performed first along y and then along x.

Key Concepts

Region of Integration

In double integration, the **region of integration** is the area over which we integrate our function. Understanding this region is crucial because it determines the limits of the integral, which in turn affects how the integral is evaluated. For the given problem, the described integral is \( \int_{0}^{3} \int_{1}^{e^{y}}(x+y) \, dx \, dy \). This implies that we're integrating over a specific area in the xy-plane.

Let's break this down:

• For each fixed \( y \) between 0 and 3, \( x \) varies from 1 to \( e^y \).
• This means that as \( y \) changes from 0 to 3, \( x \) starts at the vertical line \( x=1 \) and expands to follow the curve \( x=e^y \).
• The boundaries for \( y \) are straightforward: from \( y=0 \) to \( y=3 \).

So, our region of integration is enclosed by:

• The vertical line at \( x=1 \).
• The exponential curve \( x=e^y \).
• The horizontal lines at \( y=0 \) and \( y=3 \).

To visualize, sketching these boundaries can help. The vertical line and the curve contain the area through which we'll calculate the integral.

Order of Integration

The **order of integration** refers to the sequence in which the integrations are performed in a double integral. Many problems originally present a specific order, which relates to the limits of integration provided. In our exercise, the initial order is \( \int_{0}^{3} \int_{1}^{e^{y}} (x+y) \, dx \, dy \). Here, we integrate first with respect to \( x \), and then \( y \).

• Integrating with respect to \( x \) means, for every fixed \( y \), \( x \) varies between the limits 1 and \( e^y \).
• Once we perform this integration over \( x \), the result is integrated with respect to \( y \) from 0 to 3.

It's essential to follow this defined order because switching it without adjusting the limits can significantly alter the outcome. Learning to interpret these limits in a double integration setup is a useful skill for solving complex problems.

Reversing Integration

Sometimes reversing the **order of integration** can simplify a complicated integral. To accomplish this, we must accurately adjust the limits of integration, examining the overlaps between curves and lines that define the region. In our example, initially, we were integrating with \( \, dx \, dy \) in the order of \( x \) first. Reversing this process changes our approach.

Here's how it works:

• We observe that \( y \) varies between 0 and \( \ln{x} \) when the order is switched.
• Therefore, the new limits will have \( x \) range from 1 to \( e^3 \), indicating the expanse of the region horizontally after reversing.
• Finally, \( y \) fluctuates from 0 to \( \ln{x} \), consistent with the original integration region definition formed by the curve \( x = e^y \).

Consequently, our new integral expression becomes \( \int_{1}^{e^3} \int_{0}^{\ln{x}} (x+y) \, dy \, dx \). Adjusting the order is commonly employed to serve computational ease and often unveils alternative paths to evaluating what might seem like complex integral problems.