finding the region between two curves

[jay1234]

I know how to fin the region, but once I get to the last step I don't know what to subtract from what.

Example:
f(x)=x
f(x)=x^(2)-2x (-2x is not raised to any power, just an FYI)

I got x=0,3

now after I find the integral for f(a)-f(b)
what comes first...the 0 or 3. Because I have to substitute the x values into the antiderivitive, but I don't know if I should start with the 0 or the 3..because the answer will either be positive or negative. And I don't know which is the right answer.
So my question would be, Which x value comes first in the subtraction.

 

[skeeter]

\(\displaystyle \L A = \int_0^3 x - (x^2 - 2x) dx\)

a = 0, b = 3 ... area will be F(3) - F(0), where F is the antiderivative of the integrand.

 

[jay1234]

but the points are not given just like that. You have to solve the function to get the values of x. They are not given to me in any order. It's just two functions then I have to solve for all values of x, then I differentiate. I still don't know what order to put them.

 

[skeeter]

I don't understand your "confusion" ... the process for finding the area between the two curves is rather simple, and it does not involve differentiation.

1st ... find where the two functions intersect

\(\displaystyle \L x^2 - 2x = x\)

\(\displaystyle \L x^2 - 3x = 0\)

\(\displaystyle \L x(x - 3) = 0\)

so x = 0 and x = 3 are where they intersect.

2nd ... pick any value of x between 0 and 3, say x = 1.

evaluate both functions at x = 1 ...

y = x at x = 1 is 1

y = x² - 2x at x = 1 is -1

the function y = x is greater than the function y = x² - 2x in the interval from x = 0 to x = 3.

the area between two curves is always as follows ...

\(\displaystyle \L A = \int_a^b (top function) - (bottom function) dx\)

where a and b are the x-values of the points of intersection.

now ... find the area between the two curves using the definite integral I gave you in my previous post.