Find a Function F Such That $\mathbf { F } = \nabla f$

question 34

Short Answer

Find a function f such that $\mathbf { F } = \nabla f$ and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the given curve C. $\mathbf { F } ( x , y ) = y \mathbf { i } + ( x + 6 y ) \mathbf { j }$ C is the upper semicircle that starts at (1, 2) and ends at (5, 2).

---

Definitions:

Business Entities

Structures established by one or more individuals to conduct business or engage in activities for profit, including corporations, partnerships, and sole proprietorships.

Sole Proprietorship

A business structure owned by a single individual, where there's no legal distinction between the owner and the business.

Liable

Legally responsible or obligated to face consequences, typically in the form of compensations or reparations.

Profits

Financial gains obtained after subtracting the costs, expenses, and taxes from a business's total revenue.