math
posted by ana .
let R be the region bounded by the graphs of y = sin(pie times x) and y = x^3  4.
a) find the area of R
b) the horizontal line y = 2 splits the region R into parts. write but do not evaluate an integral expression for the area of the part of R that is below this horizontal line.
c) The region R is the base of a solid. For this solid, each cross section perpendicular to xaxis is a square. Find the volume of this solid.
d) the region R models the surface of a small pond. At all points in R at a distance x from the yaxis, the depth of the water is given by h(x)=3x. find the volume of the water in the pound.

y = sin (pi x) has to be between 1 and +1 It seems to me it only intercepts the y = x^3 4 curve once, at around x = 1.45.
Therefore I don't see how one can define a closed region R between the curves.
Something is fishy here 
I also noticed that you had posted this same question twice so far, and just like "drwls", I was confused by what "region R" you were talking about.
I was going to assume that the region was between the sine curve, the cubic and the yaxis, but that meant solving
sinx = x^3  4
which would be a major undertaking.
If that is the case, and once that intersection point between y = sinx and y = x^34 has been found, the question would not be that difficult.
Please clarify and one of us will attempt to help you out. 
I agree with Reiny, but believe he meant to write sin (pi*x) instead of sin x for one of the y functions