Today, Fri Jan 20, we discussed how to integrate the function

f[t_]:=Sqrt[1+t^2]

Integrate[f[t],t]

TrigToExp[%]

We solved the equation:

eq:=2*a*Sin[t]==2*a/3*(1+Sin[t])

Solve[eq,t]

Note that Mathematica gives all the solutions. Below we select the one in the first quadrant

Solve[eq && t>0 &&t<Pi/2,t]

Now define two quantities and integrate

r1:=2*a*Sin[t]

r2:=a*Sqrt[2*Cos[2*t]]

r1

r2

Integrate[r1^2-r2^2,t]

FullSimplify[%]

Integrate[r1^2-r2^2,{t,Pi/6,Pi/2}]

Expand[%]