Set up and evaluate a line integral to find the area of the region R bounded by the graph of $x ^ { 2 } + y ^ { 2 } = 25$
A) $\frac { 1 } { 2 } \int _ { C } x d y - y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$
B) $\frac { 1 } { 2 } \int _ { C } y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$
C) $\int _ { C } x d y = \iint _ { R } 1 d A = 10 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$
D) $\int _ { C } x d y - y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$
E) $\frac { 1 } { 2 } \int _ { C } x d y = \iint _ { R } 1 d A = 10 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$

Question

Quizplus · Accepted Answer

The correct formula for the area enclosed by a curve $C$ is given by $\frac { 1 } { 2 } \int _ { C } (x d y - y d x)$. The equation $x ^ { 2 } + y ^ { 2 } = 25$ describes a circle with radius 5, and the area of a circle is $\pi r^2$, which equals $25\pi$ for a radius of 5. Therefore, option A correctly applies the formula for the area enclosed by a curve and matches the expected area of the region $R$.

Set Up and Evaluate a Line Integral to Find the Area