Calculus C Line Integral Problem

Q1:

Given: $$\int_{c} yd s$$

$$c: x=t^{2} $$ (1)

$$y=2 t$$ (2)

$$0 \leq t \leq 3$$ (3)

Req: Line intergrad

Sol: (1) $$d s=\sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t$$

(2) $$d s=\sqrt{(2 t)^{2}+(2)^{2}} d t$$

$$d s=\sqrt{4 t^{2}+4} d t$$

$$d s=2 \sqrt{t^{2}+1} d t$$

(3) $$\int_{c} y d s=\int_{0}^{3} 2 t(2 \sqrt{t^{2}+1} d t$$

$$=\int_{0}^{3} 4 t \sqrt{t^{2}+1} d t$$

(4) $$u=t^{2}+1 \Rightarrow 2 t d t=d u$$

(5) $$\int_{1}^{10} 2 \sqrt{u} d u=\int_{1}^{10} 2 \cdot u^{\frac{1}{2}} d u$$

$$=\left [2 \cdot \frac{3}{2} \cdot u^{\frac {3}{2}} \right]_{1}^{10} $$

(6) $$\int_{c} y d s = \frac{4}{3} \cdot 10 \sqrt{10}-\frac{4}{3} \cdot 1 \sqrt {1} $$

$$=\frac{40 \sqrt{10}-4}{3}$$ #

Q2: Given:

$$\int_{c}(x+2y) d x+x^{2} d y$$

c: Line segmant 

$$(0,0) \rightarrow(2,1) \rightarrow(3,0)$$

Req: Line Integral??

Sol: (1) graph c 

(2) $$C :\left\{\begin{array}{l}{x=t ; y=\frac {1}{2}t, \quad 0<t \leq 2} \\ {x=t ; y=3-t \quad 2<t \leq 3}\end{array}\right.$$

(3) $$=\int_{0}^{2} t+2\left(\frac{1}{2} t\right) d t+t^{2}\left(\frac{1}{2} d t\right)+\int_{2}^{3} t + 2(3-t)+t^{2}(-dt)$$

(4) $$=t^{2}+\left.\frac{1}{6} t^{3}\right|_{0} ^{2}+6 t-\frac{1}{2}t^{2}-\left.\frac{1}{3} t^{3}\right|_{2} ^{3} $$

(5) $$=4+\frac{8}{6}+18-\frac{9}{2}-9-12+2+\frac{8}{3}=\frac{5}{2} $$