Math Problem Statement
∫01∫x^2,2-x^2 (xy))dydx reverse order and solve
Solution
To reverse the order of integration and solve the double integral:
Step 1: Identify the region of integration
We first need to understand the limits of the integral. The inner integral is with respect to , and it ranges from to . The outer integral is with respect to , and it ranges from to .
So the region of integration is defined by:
Step 2: Reverse the order of integration
Now we want to switch the order of integration, i.e., integrate with respect to first, then with respect to .
From the bounds, we can express the region as:
- For a fixed , the values of range from to (since ) and from to (since ).
Thus, the new integral becomes:
Step 3: Solve the integral
First part:
- Integrate with respect to :
- Now, integrate with respect to :
Second part:
- Integrate with respect to :
- Now, integrate with respect to :
Step 4: Final result
Adding both parts together:
Thus, the value of the double integral is .
Would you like a detailed explanation on any step?
Here are 5 related questions:
- How do you determine the region of integration from given bounds?
- What techniques can be used to reverse the order of integration for non-rectangular regions?
- How can substitution help simplify double integrals?
- What are common applications of double integrals in physics or engineering?
- How does changing the order of integration affect the limits?
Tip: Always sketch the region of integration when reversing the order of integration for clarity.
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Math Problem Analysis
Mathematical Concepts
Double Integrals
Reversing Order of Integration
Integration in Rectangular Regions
Formulas
∫∫_R f(x,y) dy dx
Integral bounds transformation: From ∫_0^1∫_x^2^(2-x^2) xy dydx to ∫_0^1 ∫_0^√y xy dxdy
Theorems
Fubini's Theorem
Basic Integration Theorem
Suitable Grade Level
Undergraduate Calculus
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