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

question 80

Short Answer

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

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Definitions:

Intercellular Spaces

Gaps or areas between individual cells within a tissue, which allow for the exchange of gases and the transfer of nutrients and waste.

Chloroplasts

Organelles found in plant cells and some algae that conduct photosynthesis, converting light energy, water, and carbon dioxide into oxygen and glucose.

Waxy Cuticle

A protective, hydrophobic layer found on the outer surface of plant leaves and stems, consisting mainly of waxes and lipids, which helps to reduce water loss.

Mesophyll

The inner tissue of a leaf, containing chloroplasts where photosynthesis takes place.