please show the steps Thank you Find the volume of the solid

please show the steps. Thank you

Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 4 - x2 and the plane y = 2.

Solution

Okay, so this can be solved through the use of iterated integrals which I will do below. The first quadrant is where x>0, y>0, and z>0. So this is the case when 4-x^2>0 which is true for -2<x<2, but we are only concerned for positive x so 0 <x<2. Furthermore, we must find where y>0, since the function is independent of y and is bounded by y=2 all the y can be is0<y<2.

Now, it\'s time for some integrating, because y is not a function of x, it matters not what you integrate with respect to first, so the volume is the
U(U(4-x^2,y,0,2),x,0,2) where U is the integral sign.

Integrating wrt y

= U(4y - yx^2)

apply limits of y

= U (8 - 2x^2)dx

integrating wrt x

=

8x - 2/3 x^3

apply limits of x

=

16 - 16/3

=

32/3