Questions: Compute the integral from 3 to 5 and from 3 to 4 of (x^2 - 3xy + y^2 + 3x + 5y - 4) dx dy.

Compute the integral from 3 to 5 and from 3 to 4 of (x^2 - 3xy + y^2 + 3x + 5y - 4) dx dy.
Transcript text: Compute $\int_{3}^{5} \int_{3}^{4}\left(x^{2}-3 x y+y^{2}+3 x+5 y-4\right) d x d y$.
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Solution

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Solution Steps

To solve the given double integral, we will first integrate the function with respect to \(x\) from 3 to 4, and then integrate the resulting expression with respect to \(y\) from 3 to 5. This approach involves evaluating the inner integral first and then using its result to evaluate the outer integral.

Step 1: Define the Function

The function to be integrated is given by: \[ f(x, y) = x^2 - 3xy + y^2 + 3x + 5y - 4 \]

Step 2: Compute the Inner Integral

First, we integrate \(f(x, y)\) with respect to \(x\) from 3 to 4: \[ \int_{3}^{4} (x^2 - 3xy + y^2 + 3x + 5y - 4) \, dx \] The result of this integration is: \[ y^2 - \frac{11}{2}y + \frac{113}{6} \]

Step 3: Compute the Outer Integral

Next, we integrate the result of the inner integral with respect to \(y\) from 3 to 5: \[ \int_{3}^{5} \left(y^2 - \frac{11}{2}y + \frac{113}{6}\right) \, dy \] The result of this integration is: \[ \frac{79}{3} \]

Final Answer

The value of the double integral is \(\boxed{\frac{79}{3}}\).

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