Q8Mathematics I

Question

Change the order of integration of the following double integration:

\int_{0}^{a} \int_{x}^{2 a - x} f (x, y) d y d x

Answer 1

The original integral defines a solid geometric triangle bounded perfectly by $y = x$ , $y = 2 a - x$ , and the vertical axis $x = 0$ . To reverse the order to $d x d y$ , we must slice this specific region horizontally.

Because the right-side bounding curve changes drastically at the vertex $(a, a)$ , the region must be mathematically split into a bottom half (bounded strictly by $x = y$ ) and a top half (bounded strictly by $x = 2 a - y$ ). This absolute geometric necessity creates the two separate integrals.

Q8Mathematics I

Question

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