Math Problem Statement

Sketch the region of integration and change the order of integration.

3

ln(x

f(xydy dx

0    0    f(xydx dy    

Solution

Let's first clarify the problem you're presenting. You have a double integral with two different orders of integration:

  1. First case:
    13ln(x)xf(x,y)dydx\int_{1}^{3} \int_{\ln(x)}^{x} f(x, y) \, dy \, dx

  2. Second case (reversing the order of integration):
    00f(x,y)dxdy\int_{0}^{0} f(x, y) \, dx \, dy

The second case appears incomplete or incorrectly written, so I'll focus on the first case and show how to change the order of integration, assuming the bounds from the first expression are the relevant ones.

Step 1: Identify the region of integration

From the first integral, the bounds are:

  • xx ranges from 1 to 3: 1x31 \leq x \leq 3,
  • For each xx, yy ranges from ln(x)\ln(x) to xx: ln(x)yx\ln(x) \leq y \leq x.

We can describe the region as the area under the curves y=ln(x)y = \ln(x) and y=xy = x for xx ranging from 1 to 3.

Step 2: Sketch the region of integration

The region is bounded by:

  • The curve y=ln(x)y = \ln(x), which starts at (1,0)(1, 0) and ends at (3,ln(3))(3, \ln(3)),
  • The line y=xy = x, which goes from (1,1)(1, 1) to (3,3)(3, 3),
  • Vertical lines at x=1x = 1 and x=3x = 3.

This creates a region between the curves y=ln(x)y = \ln(x) and y=xy = x for xx values between 1 and 3.

Step 3: Change the order of integration

Now, to reverse the order of integration, we need to express xx in terms of yy. From the bounds:

  • The lower bound of yy is 0 (since ln(1)=0\ln(1) = 0),
  • The upper bound of yy is 3 (since the line y=xy = x intersects x=3x = 3).

For each yy, xx ranges from:

  • The curve x=eyx = e^y (since y=ln(x)x=eyy = \ln(x) \Rightarrow x = e^y),
  • To the line x=yx = y (since y=xy = x).

Thus, the new bounds are:

  • yy ranges from 0 to 3: 0y30 \leq y \leq 3,
  • For each yy, xx ranges from eye^y to yy: eyx3e^y \leq x \leq 3.

Step 4: Rewriting the integral

The integral with reversed order of integration is:

03eyyf(x,y)dxdy\int_{0}^{3} \int_{e^y}^{y} f(x, y) \, dx \, dy

This is the new order of integration, with yy integrated first, and then xx next.


Would you like me to provide more details or explanations on any step? Here are 5 related questions to deepen your understanding:

  1. How would you compute the double integral given a specific function f(x,y)f(x, y)?
  2. Can you graph the curves y=ln(x)y = \ln(x) and y=xy = x to visualize the region better?
  3. What happens to the region of integration if the bounds of xx or yy are changed?
  4. How does changing the order of integration affect the ease of solving the integral?
  5. What is the significance of the limits 11 and 33 for xx?

Tip: When changing the order of integration, always carefully analyze how the original bounds translate to the new variables to avoid mistakes in setting up the new integral.

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Double Integrals
Change of Order of Integration
Graphing Regions of Integration

Formulas

∫ from a to b ∫ from c(x) to d(x) f(x, y) dy dx
Region transformation: x = e^y from y = ln(x)

Theorems

Fubini's Theorem
Integration Bounds Transformation

Suitable Grade Level

Undergraduate - Calculus II/Multivariable Calculus