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∬_(R)12 x-18 ydA quad R=[-1,4]×[2,3]

$\iint_{R} 12 x-18 y d A \quad R=[-1,4] \times[2,3]$

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Does (x, y) satisfy the inequality?

Full solution

Q.

\iint_{R} 12 x-18 y d A \quad R=[-1,4] \times[2,3]

Identify Limits of Integration: Identify the limits of integration for the double integral. The region R is defined by the intervals $[-1,4]$ for $x$ and $[2,3]$ for $y$ .
Set Up Double Integral: Set up the double integral $\iint_{R}(12x - 18y)\,dA$ with the given limits. The integral becomes $\int_{-1}^{4} \left( \int_{2}^{3} (12x - 18y) \,dy \right) dx$ .
Integrate with Respect to y: Integrate with respect to y first. The integral $\int_{2}^{3} (12x - 18y) dy$ becomes $12xy - 9y^2$ evaluated from $y=2$ to $y=3$ .
Plug in Limits and Simplify: Plug in the limits for $y$ and simplify. $(12x(3) - 9(3)^2) - (12x(2) - 9(2)^2)$ simplifies to $(36x - 81) - (24x - 36)$ .
Combine Like Terms: Combine like terms to get $12x + 45$ .

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