Sorry, you're so quick, I took one of your answers for another question!
Finally,
Therefore, given my scenario, I could use the integral to calculate the area under the curve described by the derivative?
Charlie.
Yes. INT f'dx is f(x). The area under the f'curve is the integral.
Thanks bob, and thanks to drwls too, in the earlier ones.
Charlie.
Yes, in your scenario, if you have a function described by its derivative, you can use integration to calculate the area under the curve.
To do this, you would need to find the antiderivative (primitive) of the derivative function. The antiderivative is the original function from which the derivative was taken.
Once you have the antiderivative, you can evaluate it over a specific interval to find the area under the curve between two points. This is done by subtracting the value of the antiderivative function at the lower endpoint of the interval from the value of the antiderivative function at the upper endpoint.
Mathematically, if f(x) is the function described by the derivative, F(x) is its antiderivative, and a and b are the lower and upper endpoints of the interval respectively, the area under the curve is given by:
Area = F(b) - F(a)
Note that if the function described by the derivative is negative over the interval, the area will be negative, indicating that the curve is below the horizontal axis.
So, finding the antiderivative and evaluating it over the desired interval will allow you to calculate the area under the curve described by the derivative.