Dr. Brajesh Kumar Jha

Double Integration in Polar Coordinates Animated Step-by-Step Solutions Example 1: Area of Circle $x^2+y^2 \le 4$ Evaluate: $$\iint_R 1 \, dA$$ ▶ Show Detailed Solution Step 1: Convert to Polar Coordinates $x = r\cos\theta$, $y = r\sin\theta$ $x^2+y^2 = r^2$ $dA = r\,dr\,d\theta$ Step 2: Limits $0 \le r \le 2$, $0 \le \theta \le 2\pi$ Step 3: Integral $$ \int_0^{2\pi}\int_0^2 r\,dr\,d\theta $$ $$ \int_0^2 r\,dr = \frac{r^2}{2}\Big|_0^2 = 2 $$ $$ \int_0^{2\pi} 2\,d\theta = 4\pi $$ Final Answer: $4\pi$ Example 2: Quarter Circle Integral Evaluate: $$\iint_R (x^2+y^2)\, dA$$ ▶ Show Detailed Solution Step 1: $x^2+y^2 = r^2$ Limits: $0 \le r \le 3$ $0 \le \theta \le \frac{\pi}{2}$ $$ \int_0^{\pi/2}\int_0^3 r^3\,dr\,d\theta $$ $$ \int_0^3 r^3 dr = \frac{81}{4} $$ $$ \frac{81}{4} \cdot \frac{\pi}{2} = \frac{81\pi}{8} $$ Final Answer: $\frac{81\pi}{8}$ Example 3: Annular Region Evaluate: $$\iint_R 1\,dA$$ ▶ ...

Limits of Functions of Several Variables Solved Examples with Collapsible Solutions The following problems illustrate different techniques used to evaluate limits of multivariable functions, including direct substitution, algebraic simplification, polar coordinates, and the path method. Example 1 $$ \lim_{(x,y)\to(0,0)} \frac{3x^2 - y^2 + 5}{x^2 + y^2 + 2} $$ Click to View Solution Since numerator and denominator are continuous at $(0,0)$, substitute directly: $$ \text{Numerator} = 5 $$ $$ \text{Denominator} = 2 $$ $$ \boxed{\text{Limit} = \frac{5}{2}} $$ Example 2 $$ \lim_{(x,y)\to(0,0)} \frac{x^2 - xy}{\sqrt{x}-\sqrt{y}} $$ Click to View Solution Factor the numerator: $$ x^2 - xy = x(x-y) $$ Use the identity: $$ x-y = (\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y}) $$ After cancellation: $$ = x(\sqrt{x}+\sqrt{y}) $$ As $(x,y)\to(0,0)$: $$ \boxed{\text{Limit} = 0} $$ Example 3 $$ \lim_{(x,y)\to(0,0)} \frac{4xy^2}{x^2+y^2} $$ Method 1: P...