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What is the area of the part of the

x y

-plane within which $x^{3} y^{2}-x^{2}...

Aug 22, 2024

Generated Graph

Solution by Steps

step 2

Factoring the expression gives us

x y^2 (x - y) + x^2 y^2 \geq 0

step 3

Analyzing the factors, we find that

x = 0

and

y \geq 0

is a solution. For 0 < x < 1/2 , we have

x \leq y \leq 1 - x

step 4

For

x \geq 1/2

, we find that

y = x

is valid. Thus, we can determine the area by integrating over the valid region

step 5

The area of the region defined by these inequalities is calculated to be

\frac{1}{2}

Key Concept

Area under curves and inequalities in the xy-plane

Explanation

The area is determined by analyzing the inequalities and finding the valid region in the xy-plane, which can be calculated through integration.