The work done by the force $\mathbf { F } ( x , y ) = 2 x y \mathbf { i } + \left( x ^ { 2 } + y ^ { 2 } \right) \mathbf { j }$ moving along $y ^ { 2 } = x$ from (0, 0) to (1, 1) is

A) $\frac { 5 } { 2 }$
B) $\frac { 8 } { 3 }$
C) $\frac { 4 } { 3 }$
D)0
E) $\frac { 4 } { 5 }$

Question

The work done by the force $\mathbf { F } ( x , y ) = 2 x y \mathbf { i } + \left( x ^ { 2 } + y ^ { 2 } \right) \mathbf { j }$ moving along $y ^ { 2 } = x$ from (0, 0) to (1, 1) is

A) $\frac { 5 } { 2 }$
B) $\frac { 8 } { 3 }$
C) $\frac { 4 } { 3 }$
D)0
E) $\frac { 4 } { 5 }$

Quizplus · Accepted Answer

The work done is the line integral of the force along the path. Parametrize the path $y^2 = x$ as $x = t^2$, $y = t$, $0 \leq t \leq 1$. The line integral becomes $\int_0^1 (2t^3 \cdot 2t + (t^4 + t^2) \cdot 1) \, dt = \int_0^1 (4t^4 + t^4 + t^2) \, dt = \int_0^1 (5t^4 + t^2) \, dt$. Evaluating this gives $\frac{5}{5} + \frac{1}{3} = \frac{4}{3}$.

The Work Done by the Force F(x,y)=2xyi+(x2+y2)j\mathbf { F } ( x , y ) = 2 x y \mathbf { i } + \left( x ^ { 2 } + y ^ { 2 } \right) \mathbf { j }F(x,y)=2xyi+(x2+y2)j

The Work Done by the Force $\mathbf { F } ( x , y ) = 2 x y \mathbf { i } + \left( x ^ { 2 } + y ^ { 2 } \right) \mathbf { j }$