Evaluate the double integral I = ∫_(0 to 1) ∫_(y to 1) e^(x^2) dx dy by reversing the order of integration.
Answer and explanation
Correct answer: C. (e - 1) / 2
- Step 1: The original region is bounded by y ≤ x ≤ 1 and 0 ≤ y ≤ 1, which represents the triangle with vertices (0, 0), (1, 0), and (1, 1).
- Step 2: Reversing the order: for fixed x from 0 to 1, y varies from 0 to x.
- Step 3: Integral becomes: ∫_(0 to 1) [∫_(0 to x) dy] e^(x^2) dx = ∫_(0 to 1) x e^(x^2) dx.
- Step 4: Let u = x^2, du = 2x dx: (1/2) ∫_(0 to 1) e^u du = (1/2) [e^1 - 1] = (e - 1) / 2.
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