Math Problem Statement
solve it
Solution
To solve this problem, we need to:
-
Interpret the given integral:
The problem involves changing the order of integration for the given double integral: -
Determine the region of integration:
- The inner integral is over from to .
- The outer integral is over from to .
The region is bounded by:
This describes a triangular region in the -plane. Let's visualize it.
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Change the order of integration:
To rewrite the integral with as the inner variable, analyze the bounds:- ranges from to (as we solve for given ).
- ranges from to .
The new integral becomes:
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Evaluate the new integral:
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First, compute the inner integral:
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Substitute into the outer integral:
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Compute the outer integral:
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Multiply by :
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Final Answer:
The value of the integral is:
Would you like a detailed step-by-step explanation or any further clarifications?
5 Related Questions:
- What are some tips for determining the bounds of integration when reversing order?
- How can we geometrically interpret the change of integration order?
- How does this technique generalize to non-rectangular regions?
- Can this method be applied in polar coordinates?
- What is the physical significance of such integrals in physics or engineering?
Tip:
Always sketch the region of integration to visualize the bounds clearly when reversing the order of integration.
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Math Problem Analysis
Mathematical Concepts
Double Integrals
Reversing Order of Integration
Integration Bounds
Formulas
Inner integral evaluation: \(\int_0^x y \, dy = \frac{x^2}{2}\)
Outer integral evaluation: \(\int_0^1 \frac{x^2}{2} \, dx = \frac{1}{6}\)
Theorems
Fubini's Theorem for Changing the Order of Integration
Suitable Grade Level
Undergraduate (Calculus I/II)
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