Evaluate the Line Integral Along the Curve C $\int _ { C } \left( 18 x ^ { 2 } + 5 e ^ { 5 y } + \frac { 1 } { z + 1 } \right) d s , C$

question 198

Multiple Choice

Evaluate the line integral along the curve C.
- $\int _ { C } \left( 18 x ^ { 2 } + 5 e ^ { 5 y } + \frac { 1 } { z + 1 } \right) d s , C$ is the path from $( 0,0,0 )$ to $( 1,1,1 )$ given by:
$C _ { 1 } : r ( t ) = t i , 0 \leq t \leq 1$
$C _ { 2 } : r ( t ) = i + t j , 0 \leq t \leq 1$
$C _ { 3 } : r ( t ) = i + j + t k , 0 \leq t \leq 1$

• A) \(43 + 5 e ^ { 5 } + \ln 2\)
• B) \(17 + e ^ { 5 }\)
• C) \(23 + \mathrm { e } ^ { 5 }\)
• D) \(48 + 6 e ^ { 5 } + \ln 2\)

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Definitions:

Relative Height

A concept in visual and spatial perception, indicating that objects placed lower in the visual field are seen as closer, while those higher appear further away.

Linear Perspective

Linear perspective is a technique in art and photography that creates the illusion of depth and distance on a flat surface, using converging lines and a vanishing point.

Principle

A fundamental truth, law, doctrine, or motivating force, upon which others are based or derived.

Relative Motion

The movement of an object in relation to another object, considering both of their movements in a specific frame of reference.