Question

Let R be a region in the xyâplane and β >0 be a constant. What is the region R that will minimize the value ofintegral(x^2 + y^2 - b^2)?

Answer 1

Let R be region in the xy-plane and β>0 be the constant. What is the region R that will minimize the value of integral (x² + y² - β²).

i.e.

`\int \int _R (x^2 +y^2 - \beta^2) \ dA`

Answer:

0

Explanation:

Given that:

`\int \int _R (x^2 +y^2 - \beta^2) \ dA`

where;

R is the region in the xy-plane.

To minimize our double integral, we have to determine the region over which the function we are integrating has a negative value.

x² + y² - β² ≤ 0 ; where β > 0 is a constant

x² + y² ≤ β² is the circle with center (0,0)

Radius "β" because R: x² + y² ≤ β²

The polar coordinates: x = rcosθ and y = rsinθ

x² + y² = r²

⇒ r limits : r = 0 → β

⇒ θ limits : r = 0 → 2π

`\iint_R(x^2+y^2-\beta^2) \ dA =\int\limits ^(2 \pi)_(\theta=0) \int \limits ^(\beta)_(r=0)( \beta^2 -\beta^2) \ rd \ rd\ \theta`

`\int \int _R (x^2 +y^2 - \beta^2) \ dA` = 0