MATH $2025$

Worksheet $6$

Due $12/11/24$

$3.$ For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals.

$($a$)$ Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$gradf$*dvec(r),$ where $f(x,y)=xy+e^x$ and $C$ is a straight line from the origin to $(2,1).$

$($b$)$ Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$gradf$*dvec(r),$ where $f(x,y)=x^{2}y-x$ and $C$ is any path from $(1,2)$ to $(3,2).$

$($c$)$ Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$gradf$*dvec(r),$ where $f(x,y,z)=cos(x)+sin(y)-xyz$ and $C$ is any path that starts at $(1,\frac{1}{2},2)$ and ends at $(2,1,-1).$

$4.($a$)$ Consider region $R$ bounded by parabolas $y=x^2$ and $x=y^2.$ Let $C$ be the boundary of $R$ oriented counterclockwise. Use Green's theorem to evaluate

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}(y+e^{\sqrt[\phantom{2}]{x}})dx+(2x+cos{y}^2)dy.$

$($b$)$ Use Green's theorem to evaluate the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}ydx-xdy,$ where $C$ is the circle $x^2+y^2=a^2$ oriented in the clockwise direction.MATH $2025$

Worksheet $6$

Due $12/11/24$

$3.$ For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals.

$($a$)$ Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$gradf$*dvec(r),$ where $f(x,y)=xy+e^x$ and $C$ is a straight line from the origin to $(2,1).$

$($b$)$ Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$gradf$*dvec(r),$ where $f(x,y)=x^{2}y-x$ and $C$ is any path from $(1,2)$ to $(3,2).$

$($c$)$ Evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$gradf$*dvec(r),$ where $f(x,y,z)=cos(x)+sin(y)-xyz$ and $C$ is any path that starts at $(1,\frac{1}{2},2)$ and ends at $(2,1,-1).$

$4.($a$)$ Consider region $R$ bounded by parabolas $y=x^2$ and $x=y^2.$ Let $C$ be the boundary of $R$ oriented counterclockwise. Use Green's theorem to evaluate

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}(y+e^{\sqrt[\phantom{2}]{x}})dx+(2x+cos{y}^2)dy.$

$($b$)$ Use Green's theorem to evaluate the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}ydx-xdy,$ where $C$ is the circle $x^2+y^2=a^2$ oriented in the clockwise direction.