12.3#9

I'm unsure what a and b are equal to. I'm pretty sure they should be a = -pi/2 and b = pi/2



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Well no, but you could think that if you didn't understand the geometry.  Did you follow the instructions and graph those curves?  Because that would have been very instructive, since r=1/(3 cos(θ)), which is the same as  r cos(θ)=1/3,  which is the same as x=1/3, which is a vertical straight line at x=1/3.  The constraint  -π/2≤θ≤π/2 means that you've got the whole line from y = -∞ to y = ∞.  You only get the part of the line inside the curve r=1,  which is the unit circle,  which has the equation x^2 + y^2 = 1, and when you substitute x=1/3,  you can get y^2 = 2/3 or y = ± √2/√3, which are a long way from ±∞.  Now that you have an x and a y, you should be able to use them to find your limits in θ.