Math Problem Statement
Sketch the region of integration and change the order of integration.
3
1
ln(x)
f(x, y) dy dx
0 0 f(x, y) dx dy
Solution
Let's first clarify the problem you're presenting. You have a double integral with two different orders of integration:
-
First case:
-
Second case (reversing the order of integration):
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:
- ranges from 1 to 3: ,
- For each , ranges from to : .
We can describe the region as the area under the curves and for ranging from 1 to 3.
Step 2: Sketch the region of integration
The region is bounded by:
- The curve , which starts at and ends at ,
- The line , which goes from to ,
- Vertical lines at and .
This creates a region between the curves and for values between 1 and 3.
Step 3: Change the order of integration
Now, to reverse the order of integration, we need to express in terms of . From the bounds:
- The lower bound of is 0 (since ),
- The upper bound of is 3 (since the line intersects ).
For each , ranges from:
- The curve (since ),
- To the line (since ).
Thus, the new bounds are:
- ranges from 0 to 3: ,
- For each , ranges from to : .
Step 4: Rewriting the integral
The integral with reversed order of integration is:
This is the new order of integration, with integrated first, and then next.
Would you like me to provide more details or explanations on any step? Here are 5 related questions to deepen your understanding:
- How would you compute the double integral given a specific function ?
- Can you graph the curves and to visualize the region better?
- What happens to the region of integration if the bounds of or are changed?
- How does changing the order of integration affect the ease of solving the integral?
- What is the significance of the limits and for ?
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
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