olds442 said:The Attempt at a Solution
when graphing it, it is even (looks like the cubic function) and would be 0...
but I am having problems convincing myself of this. i checked it out using my calculator, and it just gives back x|x|.. i know if you do the integral of abs x you would split it up into positive and negative parts and all that.. but this is obviously different since it goes below y=0...
An integral is a mathematical concept that is used to determine the area under a curve in a given interval. It is essentially the inverse operation of differentiation, which is used to find the slope of a curve at a specific point.
The absolute value function is used to ensure that the area under the curve is always positive. This is necessary because the integral of a function can be negative if the curve dips below the x-axis. Using the absolute value function eliminates this possibility.
The interval from -1 to 1 is used because it encompasses the entire domain of the absolute value function. This ensures that the entire curve is taken into account when calculating the area under it.
To solve this integral, you can break it into two separate integrals, one from -1 to 0 and one from 0 to 1. In each integral, you can use the fundamental theorem of calculus and the power rule for integration to find the antiderivative. Then, you can plug in the limits of integration and subtract the results to find the final answer.
The final result of this integral is 0. This is because the area under the curve from -1 to 0 is equal to the area under the curve from 0 to 1, but with opposite signs due to the absolute value function. Therefore, the two areas cancel each other out and the overall integral equals 0.