Questions: a) ∫∫D(1-y) dx dy, (D) x^2+y^2 ≤ 2y, y ≤ x^2, x ≥ 0; b) ∫∫D(xy)^(1/2) dx dy, (D) y ≥ x^3, y ≤ x^2, x ≥ 0;

Transcript text: a) $\iint_{D}(1-y) \mathrm{d} x \mathrm{~d} y$, (D) $x^{2}+y^{2} \leqslant 2 y, y \leqslant x^{2}, x \geqslant 0$; b) $\iint_{D}(x y)^{1 / 2} \mathrm{~d} x \mathrm{~d} y$, (D) $y \geqslant x^{3}, y \leqslant x^{2}, x \geqslant 0$;

Solution

Solution Steps

To solve these double integrals over the given regions, we need to first understand the regions of integration $ D $ for each part.

a) The region $ D $ is defined by the inequalities $ x^2 + y^2 \leq 2y $, $ y \leq x^2 $, and $ x \geq 0 $. The first inequality can be rewritten as $ x^2 + (y-1)^2 \leq 1 $, which represents a circle centered at $ (0, 1) $ with radius 1. The second inequality $ y \leq x^2 $ is a parabola opening upwards. We need to find the intersection of these regions and set up the integral accordingly.

b) The region $ D $ is defined by $ y \geq x^3 $, $ y \leq x^2 $, and $ x \geq 0 $. This region is bounded by the curves $ y = x^3 $ and $ y = x^2 $. We need to determine the points of intersection of these curves to set up the limits of integration.

Step 1: Define the Regions of Integration

For part (a), the region $ D $ is defined by the inequalities: \[ x^2 + (y - 1)^2 \leq 1, \quad y \leq x^2, \quad x \geq 0 \] This describes a circular region centered at $ (0, 1) $ with radius 1, intersected by the parabola $ y = x^2 $.

For part (b), the region $ D $ is defined by: \[ y \geq x^3, \quad y \leq x^2, \quad x \geq 0 \] This region is bounded by the curves $ y = x^3 $ and $ y = x^2 $, with intersection points at $ (0, 0) $ and $ (1, 1) $.

Step 2: Evaluate the Integral for Part (a)

The integral to evaluate is: \[ \iint_{D} (1 - y) \, dx \, dy \] After setting up the limits of integration based on the defined region, the result of the integral is: \[ \frac{4}{15} \approx 0.2667 \]

Step 3: Evaluate the Integral for Part (b)

The integral to evaluate is: \[ \iint_{D} (xy)^{1/2} \, dx \, dy \] With the limits of integration determined by the curves, the result of the integral is: \[ 0.0370370370370370 \approx 0.0370 \]