Find the work done by the force field $\mathbf { F } ( x , y ) = x \sin ( y ) \mathbf { i } + y \mathbf { j }$ on a particle that moves along the parabola $y = x ^ { 2 } \text { from } ( 1,1 ) \text { to } ( 2,4 )$
A) $\frac { ( 17 - \cos ( 1 ) + \cos ( 4 ) ) } { 2 }$
B) $\frac { ( 15 + \cos ( 1 ) - \cos ( 4 ) ) } { 2 }$
C) $\frac { ( 15 + \sin ( 1 ) - \sin ( 4 ) ) } { 2 }$
D) $\frac { ( 15 + \sin ( 1 ) - \cos ( 4 ) ) } { 2 }$
E) $\frac { ( 17 + \cos ( 1 ) - \cos ( 4 ) ) } { 2 }$

Question

Quizplus · Accepted Answer

The work done by the force field is calculated using the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$. Parameterize the path $C$ using $x = t$ and $y = t^2$ for $t$ from 1 to 2. Compute the integral $\int_1^2 (t \sin(t^2) + 2t^2) \, dt$, which evaluates to $\frac{(15 + \cos(1) - \cos(4))}{2}$.

Find the Work Done by the Force Field F(x,y)=xsin⁡(y)i+yj\mathbf { F } ( x , y ) = x \sin ( y ) \mathbf { i } + y \mathbf { j }F(x,y)=xsin(y)i+yj

Find the Work Done by the Force Field $\mathbf { F } ( x , y ) = x \sin ( y ) \mathbf { i } + y \mathbf { j }$