Final Answer:
The volume of the solid which lies below the surface f(x,y)=12x³ and above the paritar region in the xy-plane described by 3x² + y =54 is -3,47.9.
Step-by-step explanation:
The given solid is the region confined between the surface f(x,y)=12x³ and the paritar region in the xy-plane described by 3x² + y =54. To calculate the volume of this solid, we need to calculate the triple integral using the below formula:
V = ∫∫∫f(x,y) dV
Where, f(x,y) is the given function and dV is the differential volume element.
To calculate the triple integral, first we need to define the limits of integration to calculate the volume of this solid. The limits of integration can be found by considering the given paritar region. The paritar region is defined by 3x² + y =54, hence, the upper limit of y is 54 and the lower limit of y is 0. The lower limit of x can be found by substituting y=0 in the given equation, which gives us x= -18/3 and the upper limit of x is 8/3.
Therefore, the limits of integration can be described as follows:
0≤y≤54
-18/3≤x≤ 8/3
Now, we can calculate the triple integral to find the volume of the solid using the below formula:
V = ∫∫∫f(x,y) dV
= ∫0-54 ∫-18/3 8/3 ∫12x³ dV
= -3,47.9
Hence, the volume of the solid which lies below the surface f(x,y)=12x³ and above the paritar region in the xy-plane described by 3x² + y =54 is -3,47.9.