Using Green's Theorem, the line integral $\int $C $\left( x ^ { 2 } + y \right) d x$ where C is the parabola $y = 4 - x ^ { 2 }$ from (-2, 0) to (0, 4), from (0, 4) to (-2, 0), along the parabola, the line segment from (2, 0) to (-2, 0), is

A) $- \frac { 31 } { 3 }$
B) $- \frac { 32 } { 3 }$
C) $- \frac { 34 } { 3 }$
D) $- \frac { 35 } { 3 }$
E) $- \frac { 37 } { 3 }$

Question

Using Green's Theorem, the line integral

\int

C

\left( x ^ { 2 } + y \right) d x

where C is the parabola

y = 4 - x ^ { 2 }

from (-2, 0) to (0, 4) , from (0, 4) to (-2, 0) , along the parabola, the line segment from (2, 0) to (-2, 0) , is

A) $- \frac { 31 } { 3 }$
B) $- \frac { 32 } { 3 }$
C) $- \frac { 34 } { 3 }$
D) $- \frac { 35 } { 3 }$
E) $- \frac { 37 } { 3 }$

Quizplus · Accepted Answer

Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. The given line integral is $\int_C (x^2 + y) dx$, which can be related to Green's Theorem in the form $\int_C M dx + N dy$, where $M = x^2 + y$ and $N = 0$. According to Green's Theorem, this is equal to $\int\int_D (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) dA$, where D is the region enclosed by the curve C. Given that $M = x^2 + y$, we have $\frac{\partial M}{\partial y} = 1$ and $\frac{\partial N}{\partial x} = 0$, so the double integral becomes $\int\int_D (-1) dA$, which is the negative of the area of D. The area under the parabola $y = 4 - x^2$ from $x = -2$ to $x = 2$ can be found by integrating $4 - x^2$ with respect to $x$ over this interval. The area is $\int_{-2}^{2} (4 - x^2) dx = \left[4x - \frac{x^3}{3}\right]_{-2}^{2} = \left[(8 - \frac{8}{3}) - (-8 + \frac{8}{3})\right] = 16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3}$.Thus, the value of the line integral, which is the negative of this area, is $-\frac{32}{3}$.

Using Green's Theorem, the Line Integral ∫\int ∫ C (x2+y)dx\left( x ^ { 2 } + y \right) d x(x2+y)dx

Using Green's Theorem, the Line Integral $\int$ C $\left( x ^ { 2 } + y \right) d x$