Let f be the function given by f(x)=(x^3)/4 - (x^2)/3 - x/2 + 3cosx. Let R be the shaded region in the second quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and line l, the line tangent to the graph of f at x=0. Graph shown through link.

Question

Let f be the function given by f(x)=(x^3)/4 - (x^2)/3 - x/2 + 3cosx.  Let R be the shaded region in the second quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and line l, the line tangent to the graph of f at x=0.  Graph shown through link.

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1.  Find the area of R.
2.  Find the volume of the solid generated when R is rotated about the horizontal line y=-2.
3. Write, but do not evaluate, an integral expression that can be used to find the area of S.

Bot · Answer

1. The area of R is approximately 4.8.
2. The volume of the solid generated when R is rotated about the horizontal line y=-2 is approximately 28.7.
3. The integral expression that can be used to find the area of S is ∫0π (f(x) + 2) dx.