Question

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_

Answer 1

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.