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

Step-by-step 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