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)

To sketch the graph of the function f(x) = -xe^(-x^2) and find the area of the region bounded below by the graph and above by the x-axis from x = 0 to x = 1, follow these steps:

Step 1: Plotting the points

1. Choose some x-values, such as 0, 0.25, 0.5, 0.75, and 1.
2. Substitute these x-values into the function f(x) = -xe^(-x^2) to find the corresponding y-values.
3. Plot the points (x, y) on the graph paper.

Step 2: Sketching the graph

1. Connect the plotted points smoothly with a curve.
2. Ensure that the curve is below the x-axis for the given interval, as indicated by the function f(x) = -xe^(-x^2).

Step 3: Finding the area

1. Identify the region bounded by the graph and the x-axis within the interval x = 0 to x = 1.
2. Calculate the area of this region by integrating the function f(x) = -xe^(-x^2) over the given interval using definite integration.
The integral can be represented as ∫[0, 1] -xe^(-x^2) dx.
You can find this integral by using integration techniques or using software such as Wolfram Alpha or a numerical integration tool.
3. Evaluate the definite integral, which will give you the area of the region.

Once you have sketched the graph and found the area using integration, you can turn in your sketch and write the calculated area accordingly.