How do you find the limits of integration? Triple Integral?

Question

Find the limits of integration for ∫∫∫ dx dz dy for region bounded by:
In the first octant
paraboloid y = x² + 5z²
y = 4
y = 10

1) How do you generally approach these problems? I had always drawn out double integrals, but with triple integrals its not so convenient. I don't have any system or method now.

So far I have
∫[4,10]∫[0,?]∫[0,?] dx dz dy

Answer 1

I usually work outside-inward when the order of the integrals is specified like this.



Inside that outer integral (with respect to y), you can treat y as a constant. Now you have the double integral you're used to, in a plane parallel to the x-z coordinate plane a distance y away. The cross-section in that plane is a 2d region bounded by the x-axis, z-axis and the curve x² + 5z² = y.



You're integrating w.r.t. z next, so note that the maximum value of z (when x=0) is √(y/5), so your second integral is from 0 to √(y/5).



Finally, in the inner integral, both y and z are "momentarily constant" so the limits are from x=0 to √(y - 5z²).



There's not much difference working from inside outward, symbolically. It's just easier for me to mentally "reduce to a 2D problem" in the above order. You'll still treat y,z as constants for the innermost integral and y as constant for the middle integral, and get the same limits.

Answer 2

10 = (x^2) + 5*(z^2) -> x = sqrt(10 - 5*(z^2));



4 = (x^2) + 5*(z^2) -> x = sqrt(4 - 5*(z^2));



int_0^sqrt(2) int_0^sqrt(10 - 5z²) int_0^(x² + 5z²) dy dx dz -



int_0^(2/sqrt(5)) int_0^sqrt(4 - 5z²) int_0^(x² + 5z²) dy dx dz =



(5sqrt(5)π/2) - (2π/sqrt(5)) = 21π*sqrt(5)/10;



You can use Jacobin Matrix to solve this much quicker:



x = r*cos(θ); z = r*sin(θ)/sqrt(5); y = r^2;



dx dz = r/sqrt(5) dr dθ;



int_0^(π/2) int_2^sqrt(10) int_0^(r^2) (r/sqrt(5)) dy dr dθ = 21π*sqrt(5)/10;

Answer 3

x in [ 0 , √ ( y - 5 z²) ] , z in [ 0 , √ ( y / 5 ) ] , y in [ 4 , 10 ]...sketch it.....y = x² in z = 0 plane & y = 5 z² in the x = 0 plane......then in a fixed y plane you have a portion of an ellipse.......then the planes y = 4 & y = 10