P5 Integration - Is this cheating? Watch
P5 Integration Heinemann 3A Q42
"Find the area of the finite region bounded by the curve with equation
x² - y² = 4 and the line x = 5"
For backgound and 2 solutions provided by BCHL85 and Nima, see this
While previously stuck and struggling alone, I decided to try integrating the polar equation instead.
from x² + y² = r² (1)
and 2y² = 2r².sin²θ (2)
4 = x² - y² = (1)-(2) = r²(1-2sin²theta)
r² = 4 sec 2θ
½∫r² dθ = ln|sec 2θ + tan 2θ|
θ top limit is given by point where x=5 cuts the hyperbola (y=√21)
so θ = arccos (5 ÷ √46)
θ bottom limit equates to zero.
so enclosed area between 0, initial line, the curve is 3.134
Area of triangle between 0, initial line, (5,√21) is 11.456
Subtract to give 8.322, as before.
Is this cheating? If a polar version produces a better route, is it OK?
I won't say it was quicker, because I made a pig's ear of the
integration of r² to start with, and had to retrace my steps twice!
Sometimes switching coordinate systems might help a lot with integration, but I wouldn't recommend doing it on your A-level exams. You don't know how competent the person grading your paper is, so he could think you accidentally got the correct answer, and give you almost no marks for your working. I wouldn't risk it.
I felt that, in a way, I was avoiding the question!
You should be able to get credit for extra-specially clever things in P4, P5, P6 - but it's worth sticking to more obvious stuff in P3.
At least, that's what she reckons.