(a) Explain how, by observation, you know that a form of a

Chapter 16, Problem 43

(choose chapter or problem)

Evaluating a Line Integral Evaluate the line integral

\(\int_{C} y^{2} d x+2 x y d y\).

(a) \(C: \mathbf{r}(t)=(1+3 t) \mathbf{i}+(1+t) \mathbf{j}, \quad 0 \leq t \leq 1\)

(b) \(C: \mathbf{r}(t)=t \mathbf{i}+\sqrt{t} \mathbf{j}, \quad 1 \leq t \leq 4\)

(c) Use the Fundamental Theorem of Line Integrals, where C is a smooth curve from (1, 1) to (4, 2).

Text Transcription:

Int_C y^2 dx + 2xy dy

C:r(t)=(1+3t)i+(1+t)j, 0 leq t leq 1

C:r(t)=ti + sqrt t j, 1 leq t leq 4