Question

Find the area between the curves:
y=x^3−9x^2+14x
and y=−x^3+9x^2−14x

Answer 1

1. determine where the two functions equal each other; find the pertinent x value or values.
2. Expecially if you graph these 2 functions, this will tell you the limits of integration.
3. Subtract the 2nd equation from the first. Your result will be the distance between the 2 curves for any x at which this makes sense.
4. Integrate this distance formula from x=a to x=b (assuming you have already found the values of a and b in part 1).

Answer 2

Don't I find the integral of each equation? Then plug in my values of x=a and x=b into the new equation?

Answer 3

Or did you already say that? I'm sorry

Answer 4

\(\color{black}{\displaystyle 2\int_{a}^{b} \pm f(x)dx+2\int_{b}^{c} \pm f(x)dx+ ... }\)

where a, b, c (and on) are the intersections between f(x) and -f(x),
and the \(\pm\) is there, depending on which is above.

(Alternatively, you may just write
\(\color{black}{\displaystyle 2\left|\int_{a}^{b} f(x)dx\right|+2\left|\int_{b}^{c} f(x)dx\right|+ ... }\)

Answer 5

but, if you are new to this topic, then ignore me.

Answer 6

Do i add two areas because of the extra curve?

Answer 7

I asked you to graph both functions, and still recommend that you do so.

If one function is always above (higher up on) the other, then you need only subtract the lower function from the upper once.

If, however, the lower function becomes the upper and vice versa, then you'll need to evaluate 2 separate integrals.


Here's my original response to you:

1. determine where the two functions equal each other; find the pertinent x value or values.
2. Expecially if you graph these 2 functions, this will tell you the limits of integration.
3. Subtract the 2nd equation from the first. Your result will be the distance between the 2 curves for any x at which this makes sense.
4. Integrate this distance formula from x=a to x=b (assuming you have already found the values of a and b in part 1).