Evaluate the heat integral $$int_C (x+2y)dx + x^2dy,$$ where $C$ is composed of line segments from $(0,0)$ to $(2,1)$ and from $(2,1)$ come $(3,0)$.

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How perform you solve this by utilizing the following parametrics? I split them up yet got a an adverse answer that -1/3. What"s wrong?

For $C_1$ got, $langle t, t/2 angle$, $0 leq t leq 2$.

For $C_2$ got, $langle t, 3-y angle$, $2 leq t leq 3$.


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Hints:

$$;;;(0,0) o(2,1),:;;; 0le xle 2;,;;y=frac x2implies$$

$$intlimits_(0,0)^(2,1)(x+2y)dx+x^2dy=intlimits_0^2 (x+x)dx+x^2left(frac12,dx ight)=intlimits_0^2left(frac12x^2+2x ight)dx=$$

$$=frac16cdot8+4=frac163$$

and something similar with the other one...

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Added:

$$(2,1) o(3,0):;;;2le xle 3;,;;y=-x+3implies$$

$$intlimits_(2,1)^(3,0(x+2y)dx+x^2dy=intlimits_2^3 (x+2(-x+3))dx+x^2left((-1),dx ight)=intlimits_2^3left(-x^2-x+6 ight)dx=$$

$$left.-frac13x^3 ight|_2^3-left.left.frac12x^2 ight|_2^3+6x ight|_2^3=-frac193-frac52+6=-frac176$$


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edited Apr 30 "13 in ~ 21:05
reply Apr 30 "13 at 20:56
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DonAntonioDonAntonio
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Try writing these integrals as$$int_C (x+2y,x^2)cdot (dx,dy)$$where $cdot$ is the usual inner product.Now, for $C_1$, make $(x,y)=(x(t),y(t))=(t,fract2)$ (if your parametrization is correct), indigenous which we have $dx=dt$ and also $dy=fracdt2$. You have the right to now easily integrate with respect come $t$, within the range where $t$ varies.


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answered Apr 30 "13 in ~ 20:50
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MarraMarra
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