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)

Note: People have been telling me that the two lines do not intersect but they do there has to be an answer. please help me.
Thanks

graph of your function

http://www.wolframalpha.com/input/?i=y%3D++-xe%5E%28-x%5E2%29++from+0+to+1

notice the region will be below the x-axis, so
height = 0 - (-xe^(-x^2))
= xe^(-x^2)

area = integral (xe^(-x^2)) dx from 0 to 1
= (-1/2) e^(-x^2) from 0 to 1
= (-1/2) e^-1 - (-1/2) e^0
= -1/(2e) + 1/2
= (1/2) ( 1 - 1/e) = appr .316

Thanks for your help and for the steps. It helped me a lot.

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:

1. Graphing the function:
a. Start by choosing some x-values within the interval [0, 1]. For example, you can choose x = 0, x = 0.5, and x = 1.
b. Substitute these x-values into the function f(x) = -xe^(-x^2) to find the corresponding y-values.
- For x = 0: f(0) = -(0)e^(0) = 0
- For x = 0.5: f(0.5) = -(0.5)e^(-0.5^2) = -0.5e^(-0.25)
- For x = 1: f(1) = -(1)e^(-1^2) = -e^(-1)
c. Plot the points (0, 0), (0.5, -0.5e^(-0.25)), and (1, -e^(-1)) on a graph paper.
d. Connect these points with a smooth curve, taking into consideration the overall shape of the function.

2. Finding the area:
a. From the graph, you can see that the region bounded below by the graph and above by the x-axis lies between x = 0 and x = 1.
b. You need to calculate the area of this region. Since the function is negative, we need to consider the absolute value of the function.
c. The formula to find the area under a curve within an interval is given by the definite integral: A = ∫ |f(x)| dx.
d. The integral of the function |-xe^(-x^2)| from x = 0 to x = 1 gives you the area of the region.
e. Substitute the limits and integrate:
A = ∫ |f(x)| dx from 0 to 1
= ∫ |-xe^(-x^2)| dx from 0 to 1
f. Unfortunately, finding an analytical solution for this integral is not straightforward. You may need to use numerical methods or calculators that can perform definite integrals to find the approximate area.

While it is difficult to obtain a precise answer without performing the integration, by following these steps, you can sketch the graph and explain the process of finding the area of the region bounded by the function and the x-axis.