Evaluate the Line Integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$

question 113

Multiple Choice

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where F $( x , y ) = 2 x y \mathbf { i } + x ^ { 2 } \mathbf { j } + \mathbf { k }$ , and the curve $C$ is given by the vector function $r ( t ) = t ^ { 2 } \mathbf { i } + t ^ { 3 } \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 1$ .

• A) \(\frac { 1 } { 4 }\)
• B) \(\frac { 1 } { 2 }\)
• C)\(\frac { 3 } { 4 }\)
• D) \(1\)
• E) \(\frac { 5 } { 4 }\)
• F)\(\frac { 3 } { 2 }\)
• G)\(\frac { 7 } { 4 }\)
• H) \(2\)

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Definitions:

Permanent Magnet Motors

Electric motors that utilize permanent magnets to generate the magnetic field within the motor, rather than an electrical coil.

Field-wound Motors

Electric motors where the magnetic field is generated by winding wire around a core to create electromagnets, typically adjustable for speed control.

Efficiency

The ratio of useful output of a system to its total input, often expressed as a percentage, indicating how well the system converts input energy into output work or desired form of energy.

Series Motor

A motor that contains only a series field winding.