Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( e ^ { x ^ { 2 } } - y ^ { 3 } , \cos y ^ { 2 } + x ^ { 3 } \right)$ and the curve C is given by C : $x = 2 \sin t , y = 2 \cos t , 0 \leq t \leq 2 \pi$ .
A) $24 \pi$
B)$12 \pi$

C)$48 \pi$
D) $96 \pi$
E) $- 24 \pi$
F)$- 12 \pi$
G) $- 48 \pi$

H)$0$

Question

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( e ^ { x ^ { 2 } } - y ^ { 3 } , \cos y ^ { 2 } + x ^ { 3 } \right)$ and the curve C is given by C : $x = 2 \sin t , y = 2 \cos t , 0 \leq t \leq 2 \pi$ .
A) $24 \pi$
B)$12 \pi$
 
C)$48 \pi$ 
D) $96 \pi$ 
E) $- 24 \pi$ 
F)$- 12 \pi$ 
G) $- 48 \pi$

H)$0$

Quizplus · Accepted Answer

To evaluate the line integral $\int _ { C } \mathbf { F } d \mathbf { r }$, we substitute the parametric equations $x = 2 \sin t$ and $y = 2 \cos t$ into $\mathbf { F } ( x , y )$, and then use the fact that $dx = 2 \cos t dt$ and $dy = -2 \sin t dt$. The integral becomes:$$\int_{0}^{2\pi} \left( e^{(2\sin t)^2} - (2\cos t)^3 \right) 2\cos t dt + \left( \cos(2\cos t)^2 + (2\sin t)^3 \right) (-2\sin t) dt$$Simplifying and integrating term by term, we notice that the terms involving $e^{4\sin^2 t}$ and $\cos(4\cos^2 t)$ do not contribute to a net change over one period $0 \leq t \leq 2\pi$, due to their periodic nature and symmetry. The terms that contribute are those involving powers of sine and cosine, specifically the $8\sin^3 t \cos t$ and $8\cos^3 t \sin t$ terms, which, when integrated over one period, result in a net contribution of $24\pi$. This is because the integral of $\sin^3 t \cos t$ and $\cos^3 t \sin t$ over one period yields a non-zero value that, when multiplied by the coefficients and combined, results in $24\pi$.

Evaluate ∫CFdr\int _ { C } \mathbf { F } d \mathbf { r }∫C​Fdr

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$