Problem 45 - Thomas Calculus

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

Step 1: Understand the Initial Region of Integration

The given double integral is \( \int_{0}^{3} \int_{1}^{e^{y}}(x+y) \, dx \, dy \). Here, we integrate first with respect to \(x\) from \( x = 1 \) to \( x = e^y \), and then \( y \) from \( y = 0 \) to \( y = 3 \). This defines a region in the xy-plane.

Step 2: Sketch the Region

To sketch the region, consider the bounds for \( x \) and \( y \):- \( x = 1 \): a vertical line- \( x = e^y \): a curve that starts at \( x = 1 \) when \( y = 0 \) and increases as \( y \) increases- \( y \) ranges from \( 0 \) to \( 3 \), limiting the area vertically.This describes a region under the curve \( x = e^y \) and above the line \( x = 1 \).

Step 3: Determine the New Bounds for x and y

To reverse the order of integration, we need the bounds for \( y \) in terms of \( x \). From \( x = e^y \), rearranging gives \( y = \ln(x) \). Since \( y \) goes from 0 to 3, \( x \) varies from when \(x = e^0 = 1\) to \(x = e^3\). Thus, the integration for \( y \) now goes from \( 0 \leq y \leq \ln(x) \) while \( x \) runs from \( 1 \) to \( e^3 \).

Step 4: Rewrite the Integral with Reversed Order

Using the newly determined bounds, the integral with reversed order is:\[\int_{1}^{e^3} \int_{0}^{\ln(x)} (x+y) \, dy \, dx\]This accounts for \( y \) ranging from \( 0 \) to \( \ln(x) \) for each \( x \) between \( 1 \) and \( e^3 \).

Key Concepts

Order of Integration

When dealing with double integrals, the concept of 'order of integration' refers to the sequence in which you perform the integration with respect to each variable. In the given exercise, the initial order of integration is to integrate first with respect to \( x \) and then \( y \). This means that for each fixed \( y \), you compute the integral over \( x \).
Understanding and manipulating the order is valuable because reversing the order can simplify the computation, especially if the region of integration contains curves that are difficult to describe with one order but easy with another. Reversing this order changes the complexity and parameters of the function being integrated, as seen in the step-by-step solution.
In our case, initially integrating over \( x \) involves limits from \( x = 1 \) to \( x = e^y \) and then \( y \) from \( 0 \) to \( 3 \). When the order is reversed, the integration order changes to first considering \( y \) with limits from \( 0 \) to \( \ln(x) \) and then \( x \) from \( 1 \) to \( e^3 \). Understanding these changes helps in grasping the flexibility and depth of calculus.

Region of Integration

The region of integration is the area over which you are performing your double integral. In graphical terms, it’s the shape that defines the bounds of your integration.
For the given exercise, the region is determined by the functions and limits of integration: \( x = 1 \), \( x = e^y \), and \( y = 0 \) to \( y = 3 \).

• The line \( x = 1 \) is a vertical boundary on the left.
• The curve \( x = e^y \) represents the right boundary, which varies depending on \( y \).
• The horizontal boundaries are dictated by \( y = 0 \) at the bottom and \( y = 3 \) at the top.

This defines a region beneath the curve \( x = e^y \) and above \( x = 1 \), extending from \( y = 0 \) to \( y = 3 \).

To visualize, draw the curve \( x = e^y \) which moves upwards and to the right. The region lies between this curve and the vertical line \( x = 1 \), confined vertically between \( y = 0 \) and \( y = 3 \). Sketching this helps in comprehending the area you're calculating with the double integral.

Integration Bounds

Integration bounds specify the limits for each variable's range within the double integral. Initially, these bounds were set for \( x \) from \( 1 \) to \( e^y \) and \( y \) from \( 0 \) to \( 3 \).
Reversing the order of integration, as required in this exercise, necessitates finding the bounds in terms of the other variable.

• Think of the bounds for \( y \) as dependent on \( x \), instead of \( x \) on \( y \).
• The curve \( x = e^y \) can be rewritten as \( y = \ln(x) \).
• Given \( y \) ranges from \( 0 \) to \( 3 \), \( x \) ranges from \( e^0 = 1 \) to \( e^3 \).

Thus, the new integration bounds for \( y \) are \( 0 \) to \( \ln(x) \), while for \( x \) they adjust to \( 1 \) to \( e^3 \).

These steps demonstrate how changing bounds is critical for successfully reversing the order of integration. Each set of bounds dictates the limits and direction for exploring the integral's region, crucial for accurate calculations.