Question

5. Let $F(x, y) = e^yi + xe^yj$ be defined on a smooth curve $C$ given by the vector function $r = ti + t^2j$ for $0 \le t \le 1$. Prove that $\int_C F \cdot dr = e$ using the fundamental theorem of calculus for vector fields.

          5. Let $F(x, y) = e^yi + xe^yj$ be defined on a smooth curve $C$ given by the vector
function $r = ti + t^2j$ for $0 \le t \le 1$. Prove that
$\int_C F \cdot dr = e$
using the fundamental theorem of calculus for vector fields.

5. Let F(x, y) = e^yi + xe^yj be defined on a smooth curve C given by the vector
function r = ti + t^2j for 0 ≤ t ≤ 1. Prove that
F · dr = e
using the fundamental theorem of calculus for vector fields.

Added by Felipe A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Ben Blakesley

Step 1

The line integral of F along C is given by: ∫ F · dr = ∫ (ei + ej) · (ri + rj) dt = ∫ (ti + tj) · (ei + ej) dt = ∫ (tei + tej + tei + tej) dt = ∫ (2tei + 2tej) dt = 2 ∫ (tei + tej) dt = 2 ∫ (ti + tj) dt = 2 ∫ (r) dt = 2 ∫ (ti + tj) dt = 2 ∫ (ri + rj) dt = 2 ∫ (ri Show more…

Show all steps

Thanks for your feedback!

Let F = ei + ej be defined on a smooth curve C given by the vector function r = ti + tj for 0 < t < 1. Prove that, using the fundamental theorem of calculus for vector fields.