Find a region R such that double integral $\int _ { R } \left( 2 - x ^ { 2 } - y ^ { 2 } \right) d A$ has the largest value.
A) $x ^ { 2 } - y ^ { 2 } \leq 2$
B) $x ^ { 2 } - y ^ { 2 } \geq 2$
C) $x ^ { 2 } + y ^ { 2 } \leq 2$
D) $x ^ { 2 } + y ^ { 2 } \geq 2$

Question

Quizplus · Accepted Answer

The integrand $2 - x^2 - y^2$ is positive when $x^2 + y^2 < 2$ and becomes zero when $x^2 + y^2 = 2$. Thus, the integral has the largest value when the region of integration is within the circle $x^2 + y^2 \leq 2$, which maximizes the area where the integrand is positive.

Find a Region R Such That Double Integral ∫R(2−x2−y2)dA\int _ { R } \left( 2 - x ^ { 2 } - y ^ { 2 } \right) d A∫R​(2−x2−y2)dA

Find a Region R Such That Double Integral $\int _ { R } \left( 2 - x ^ { 2 } - y ^ { 2 } \right) d A$