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Piece$-$wise continuous polynomial curves known as splines are frequently used to provide a math$-$

ematical description of engineered surfaces. Splines using third$-$order polynomials are the most

common, known as cubic splines, by using a spline in the x dimension and the y dimension we can

describe surfaces. The spline surface defined by the equation:

$z(x,y)=x{y}^3+y^3+x+1$

is shown here:

$($the file generating this figure is on blackboard if you'd like to zoom or rotate it around to help

you understand the problem$)$ Suppose this surface has a uniform thickness of $1$ mm and is

composed of inconel $718$ which has a density of $8200kg{m}^{-3}.$ This part is defined on the interval

$[-1,1]m$ along both the x and y dimensions. Find an expression for the mass of this part.

Note that you may not be able to fully finish the integration by hand. Fully define the integral to

be computed however $($limits$,$ integrand, variables to integrate over$).$ What can you conclude

about integrating a scalar over a surface?