Sketch the graph (Do this on paper. Your teacher may ask you to turn in this graph.) and find the area of the region bounded below by the graph of the function and above by the x axis from x = 0 to x = 1.

f(x) = -xe^(-x^2)

Scroll down a ways and see all the problems posted by "Vanessa" which have already been solved.

Or, look at the related questions, where I think you will find links to them.

To sketch the graph of the function f(x) = -xe^(-x^2), we can follow these steps:

1. First, plot some key points to get an idea of the behavior of the function:
- When x = 0, f(0) = 0.
- As x approaches infinity, the function approaches zero.
- As x approaches negative infinity, the function also approaches zero.
- As x becomes positive and larger, the function becomes more negative.

2. Next, find the x-intercepts of the function by setting f(x) = 0 and solving for x:
- -xe^(-x^2) = 0
- This equation has one real solution at x = 0.

3. Now, compute the behavior of the function for some selected values of x between -1 and 1. This will give us an idea of how the function behaves in that interval. For example:
- When x = 1, f(1) = -e^(-1) ≈ -0.368.
- When x = -1, f(-1) = -e^(-1) ≈ -0.368.

4. With all these points and behavior information, sketch the graph on paper. Be sure to include the x-axis and label the key points.

To find the area of the region bounded below by the graph of the function and above by the x-axis from x = 0 to x = 1, we can use definite integration. Since the function is negative in this region, we need to take the absolute value of the function before integrating.

The integral representing the area can be calculated as follows:

∫[0, 1] |-xe^(-x^2)| dx

Simplifying this integral:

∫[0, 1] xe^(-x^2) dx

You can evaluate this integral using various integration techniques such as substitution or integration by parts, depending on your knowledge or the techniques taught by your teacher. After evaluating the integral, you will find the area of the region bounded by the graph of the function and the x-axis from x = 0 to x = 1.