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How to solve integrals? \( \iint_{D}(x+y+1) d \sigma \), where \( D \)-rectangle, \( 0 \leq x \leq 1,0 \leq y \leq 2 \). \( \frac{x^{2}}{4}+\frac{y^{2}}{9} \leqslant 1 \), fud the Boundaiies of the double mtegral? \[ \iint_

User Herve
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Final Answer:

The boundaries for the double integral
\( \iint_(D)(x+y+1) \, d\sigma \),where
\( D \) is a rectangle defined by
\( 0 \leq x \leq 1 \) and
\( 0 \leq y \leq 2 \), subject to the constraint
\( (x^(2))/(4) + (y^(2))/(9) \leq 1 \), are
\( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 3\sqrt{1 - (x^(2))/(4)} \). These boundaries encompass both the rectangle
\( D \)and the ellipse defined by the constraint, ensuring a comprehensive integration over the specified region.

Explanation:

To find the boundaries of the double integral
\( \iint_(D)(x+y+1) \, d\sigma \), where
\( D \) is defined by the rectangle
\( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 2 \), and the constraint
\( (x^(2))/(4) + (y^(2))/(9) \leq 1 \), follow these steps:

1. Setup the Integral:

The given double integral can be expressed as:


\[ \iint_(D)(x+y+1) \, d\sigma \]

where
\( d\sigma \) represents the area element.

2. Determine Boundaries from the Rectangle:

For the rectangle
\( D \), the boundaries are
\( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 2 \).

3. Incorporate the Constraint from the Ellipse:

The constraint
\( (x^(2))/(4) + (y^(2))/(9) \leq 1 \) introduces an ellipse. To find the boundaries imposed by the ellipse, set the equation equal to 1 and solve for
\( y \):


\[ (x^(2))/(4) + (y^(2))/(9) = 1 \]

Solving for
\( y \):


\[ (y^(2))/(9) = 1 - (x^(2))/(4) \]


\[ y^(2) = 9 - (9)/(4)x^(2) \]


\[ y = 3\sqrt{1 - (x^(2))/(4)} \]

4. Combine Boundaries:

The final boundaries for
\( y \) within the ellipse are
\( 0 \leq y \leq 3\sqrt{1 - (x^(2))/(4)} \).

Therefore, the complete set of boundaries for the double integral is
\( 0 \leq x \leq 1 \) for the rectangle
\( D \), and
\( 0 \leq y \leq 2 \) for the rectangle
\( D \), with
\( 0 \leq y \leq 3\sqrt{1 - (x^(2))/(4)} \) imposed by the ellipse. These boundaries ensure that the integration is performed over the specified region.

User Targhs
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