Evaluate the line integral of a vector field over a closed triangular path by identifying it as a conservative field.

Line Integral of a Conservative Vector Field Over a Triangle

Multivariable calculus warmup

Calculate the line integral \( \int_{C} \mathbf{F} \cdot d \mathbf{r} \), where \( \mathbf{F} \) is the vector field \( \langle y^{2}, 2xy+1 \rangle \) and \( C \) is a triangle with vertices \( (0,0) \), \( (4,0) \), and \( (0,5) \), oriented counterclockwise.

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What you need first

Partial derivatives
Vector fields
Line integrals
Closed curves

What this problem tests

Tests the ability to recognize a conservative vector field and apply the property that its integral over a closed loop is zero, avoiding unnecessary parameterization.

This problem is typical in a multivariable calculus course when introducing path independence, conservative vector fields, and Green's Theorem.

Line Integral of a Conservative Vector Field Over a Triangle

What you need first

What this problem tests

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