HW 2 - Solutions

We wish to find the area of the graph of the function f(x,y)=cos((x^2)+(y^2)) over the disk 0≤(x^2)+(y^2)≤R.

Steps 1 and 2

We first parametrize and plot the surface

T[r_,θ_] = {r*Cos[θ],r*Sin[θ],Cos[(r^2)]};
ParametricPlot3D[T[r,θ],{r,0,(Sqrt[Pi/2])},{θ,0,2Pi}]

Steps 3 and 4

We’ll need to integrate ‖(Subscript[T, r])×(Subscript[T, θ])‖.

TrTθ = Cross[D[T[r,θ],r],D[T[r,θ],θ]];
Simplify[(Sqrt[TrTθ.TrTθ])]

Thus, our integral is

(∫_0^(2π))(∫_0^R)(Sqrt[(r^2)+2(r^4)-2(r^4)cos(2(r^2))])drdθ = 2π(∫_0^R)(Sqrt[(r^2)+2(r^4)-2(r^4)cos(2(r^2))])dr.

Step 5

Finally, we do it!

answer[R_]:= 2Pi*NIntegrate[(Sqrt[(r^2)+2 (r^4)-2 (r^4) Cos[2 (r^2)]]),{r,0,R}];

If R=(Sqrt[π/2]), we get

answer[Sqrt[Pi/2]]

If R=π/2, we get

answer[Pi/2]

If R=(((π/2))^2), we get

answer[(Pi^2)/4]

If R=1, we get

answer[1]

Finally, and most confusingly, if R=((π/2)^(1/4)), we get

answer[(((Pi/2))^(1/4))]