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Homework Statement
I ran into this tough integral. I am asked to compute the volume of the following region using a double integral with ##a, b, c > 0##.
The first octant region bounded by the co-ordinate planes ##x=0##, ##y=0##, ##z=0## and the cylinders ##a^2y = b(a^2-x^2)## and ##a^2z = c(a^2-x^2)##.
Homework Equations
$$V = \int \int_R f(x,y) dA$$
The Attempt at a Solution
They want this solved as a Cartesian integral, so the first thing I did was find ##z##:
$$a^2z = c(a^2-x^2) \Rightarrow z = c(1 - \frac{x^2}{a^2})$$
Hence we want the volume integral:
$$V = \int \int_R f(x,y) dA = \int \int_R c(1 - \frac{x^2}{a^2}) dA$$
Now I'm not entirely certain what the proper limits are in terms of ##x## and ##y##. If we consider the other cylinder ##a^2y = b(a^2-x^2)## with ##y=0##, we get:
$$0= b(a^2-x^2) \Rightarrow x = \pm a$$
If ##x = 0##, then:
$$a^2y = ba^2 \Rightarrow y = b$$
So I get the limits ##0 ≤ x ≤ a## and ##0 ≤ y ≤ b##. Are these limits reasonable?
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