How to determine area between 3 functions?

Question

I know how to find the area, but i need help in finding the upper and lower limit? any ideas? please explain thank u very much.
y=x+6
y= -2x
y= x^3

Answer 1

The area will be determined by sum of two integrals. First graph these functions. The point of intersection of
y=x^3 and y=x+6 is x=2,
of y= -2x and y=x+6 is x=-2, and
of y=x^3 and y=-2x is x=0. So area of the region bounded
inside the three curves will be
int (-2 to 0) [(x+6)-(-2x)] dx + int (0 to 2)[(x+6)- (x^3)] dx
= int (-2 to 0) [3x+6] dx + int (0 to 2)[x+6- x^3] dx
I hope you can evlauate these trivial integrals

Answer 2

First, graph the functions, either on paper or on a calculator, just so you get a rough idea of where the curves cross one another. Then, set the equations equal to each other so you can find the exact points of intersection. If you have the graph in front of you with the exact points, you should have no trouble determining the limits of integration. Keep in mind that because you have more than 2 functions, you may need to break up the integral into two separate integrals because you integrate "top minus bottom," but this may vary over the entire region of enclosure.

Answer 3

First of all you need to do a rough sketch of the 3 curves/lines.

The first two intersect at (-2, 4)
The first and third intersect at (2,8)
These are the points of intersection that concern us, as your diagram will show.

So you have two integrals to perform:
1) Integral from x = -2 to x = 0 of
(x+6) - (-2x)
2) Integral from x = 0 to x = 2 of
(x+6) - (x^3)

Then add them up to get your answer, which should be 6 + 10 = 16

Answer 4

You need to find the points where the functions intersect. Set them equal to each other and solve for X. I suggest graphing them also. Then you can integrate the region using the proper limits.

x+6 = -2x
x+6 = x^3
-2x = x^3
solve all three equations for x and then find y values also

Answer 5

triple integral