Answer to Question #269465 in Calculus for Margreta

Question #269465

Evaluate the line integral

∫𝒖(𝑥, 𝑦, 𝑧) × ⅆ𝒓 ,

where 𝒖(𝑥, 𝑦, 𝑧) = (𝑦^2 , 𝑥, 𝑧) and the curve 𝑪 is described by 𝒛 = 𝑦 = 𝑒 𝑥 with 𝑥 ∈ [0,1]. 



1
Expert's answer
2021-11-24T17:22:21-0500

u(x,y,z)dr=Cy2dx+xdy+zdz\int {u(x,y,z)dr = } \int\limits_C {{y^2}} dx + xdy + zdz

y=exdy=exdxy = {e^x} \Rightarrow dy = {e^x}dx

z=exdz=exdxz = {e^x} \Rightarrow dz = {e^x}dx

Then

Cy2dx+xdy+zdz=01e2xdx+xexdx+exexdx=012e2xdx+01xexdx=e2x01+01xexdx=e2e0+01xexdx\int\limits_C {{y^2}} dx + xdy + zdz = \int\limits_0^1 {{e^{2x}}dx + x{e^x}} dx + {e^x} \cdot {e^x}dx = \int\limits_0^1 {2{e^{2x}}dx + } \int\limits_0^1 {x{e^x}} dx = \left. {{e^{2x}}} \right|_0^1 + \int\limits_0^1 {x{e^x}} dx = {e^2} - {e^0} + \int\limits_0^1 {x{e^x}} dx

Since

01xexdx=u=xdv=exdxdu=dxv=ex=xex0101exdx=xex01ex01=e0e+e0=1\int\limits_0^1 {x{e^x}} dx = \left| {\begin{matrix} {u = x}&{dv = {e^x}dx}\\ {du = dx}&{v = {e^x}} \end{matrix}} \right| = \left. {x{e^x}} \right|_0^1 - \int\limits_0^1 {{e^x}dx = } \left. {x{e^x}} \right|_0^1 - \left. {{e^x}} \right|_0^1 = e - 0 - e + {e^0} = 1

Then

Cy2dx+xdy+zdz=e2e0+01xexdx=e21+1=e2\int\limits_C {{y^2}} dx + xdy + zdz = {e^2} - {e^0} + \int\limits_0^1 {x{e^x}} dx = {e^2} - 1 + 1 = {e^2}

Answer: e2{e^2}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment