Q7Mathematics I

Question

Evaluate $\iint x y d x d y$ , where the region of integration is $x + y < 1$ in the positive quadrant.

Answer 1

The integration region is a rigid right triangle located entirely in the first Cartesian quadrant, strictly bounded by the geometric line $y = 1 - x$ . We structure the iterated double integral precisely:

I = \int_{0}^{1} (\int_{0}^{1 - x} x y d y) d x

Integrating the inner layer with respect to $y$ yields $\frac{1}{2} x (1 - x)^{2}$ . We algebraically expand this polynomial to $x - 2 x^{2} + x^{3}$ and subsequently integrate with respect to $x$ across the unit interval to rigorously arrive at $\frac{1}{24}$ .

Q7Mathematics I

Question

Answer