Estimation of the volume of a solid.

[2dvh]

Homework Statement

Estimate the volume of the solid whose base is bounded by the graphs of y = x + 3 and
y = x2 − 3, and bounded by the surface z=x2+y from above, using the indicated rectangular cross sections taken perpendicular to the x-axis (figure is not the actual graph of the functions given above. It is solely a demonstration to help with the problem).

Homework Equations

The Attempt at a Solution

I've been trying to figure out a way to solve it for hours.
The only thing that I've done was figure out the area of the base, but i don't know if I am on the right track or not, seeing as the professor has not gone over this.

 

[HallsofIvy]

Did the problem really say "estimate"? It's a pretty straight forward Calculus problem to find the exact volume.
The base graphs y= x+ 3 and y= x^2- 3 (note that "x^2" is clearer than "x2") intersect where y= x+ 3= x^2-3 or x^2- x- 6= (x- 3)(x+ 2) or x= -2, y= 1 and x= 3, y= 6. Rectangular cross-sections perpendicular to the x-axis" means that the area is "height times base" where the base is the difference in y values, x+3-(x^2- 3)= -x^2+ x+ 6 and the height is $$z= x^2+ y$$. The volume of a thin "slab" would be that area multiplied by thickness, dx. Integrate that from x= -2 to 3.

 

[2dvh]

So then would the actual area end up being:
-x^4+x^3+6x^2-x^2y+xy+6y
Or am i wrong?

If it is right,would the volume be (125/12)*(2y+3)