Evaluate the Line Integral Along the Curve C $\int _ { C } ( x y + y z ) d s , C$

question 192

Multiple Choice

Evaluate the line integral along the curve C.
- $\int _ { C } ( x y + y z ) d s , C$ is the path from $( 1,1,0 )$ to $\left( e ^ { 2 } , e ^ { 2 } , 1 \right)$ given by:
$C _ { 1 } : \mathbf { r } ( \mathrm { t } ) = \mathrm { e } ^ { 2 } \mathrm { t } _ { \mathbf { i } } + \mathrm { e } ^ { 2 } \mathrm { t } \mathbf { j } , 0 \leq \mathrm { t } \leq 1$
$C _ { 2 } : \mathbf { r } ( \mathrm { t } ) = \mathrm { e } ^ { 2 } \mathbf { i } + \mathrm { e } ^ { 2 } \mathbf { j } + 9 \mathrm { t } \mathbf { k } , 0 \leq \mathrm { t } \leq 1$

• A) \(\frac { \sqrt { 2 } } { 3 } \left( e ^ { 6 } - 1 \right)\)
• B) \(\frac { \sqrt { 2 } } { 3 } \left( e ^ { 6 } - 1 \right) + 9 e ^ { 4 } + 81 e ^ { 2 }\)
• C) \(\frac { \sqrt { 2 } } { 3 } \left( e ^ { 6 } - 1 \right) + 9 e ^ { 4 } + \frac { 81 } { 2 } e ^ { 2 }\)
• D) \(\frac { \sqrt { 2 } } { 3 } \left( e ^ { 6 } - 1 \right) + e ^ { 4 } + \frac { 1 } { 2 } e ^ { 2 }\)

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Definitions:

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A process by which individuals learn to replicate or avoid behaviors based on the observed consequences of those behaviors for someone else.

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A problem-solving method involving repeated, varied attempts until success, or a solution is achieved.

Insight Learning

A form of learning characterized by the sudden realization of the solution to a problem, often without reliance on trial-and-error.