Sketch the graph and find the area of the region described below.

f(x)= -3xe^((-x)^2)
Find the area of the region bounded below by the graph of f(x) and above by the x-axis from x = 0 to x = 3.

Did you mean it as typed? because (-x)^2 is the same as (x^2) I get the feeling you meant

∫[0,3] 0-(-3xe^(-x^2)) dx
Let u = x^2 so
du = -2x dx and you have

∫[0,9] -3/2 e^-u du

Now it's a cinch, eh?

To find the area of the region described, we first need to sketch the graph of the function f(x) = -3xe^((-x)^2).

Step 1: Evaluate the function at different x-values to get some points on the graph.
By substituting various x-values into the function, we can find corresponding y-values. Let's calculate some values:

When x = 0: f(0) = -3(0)e^((0)^2) = 0
When x = 1: f(1) = -3(1)e^((-1)^2) ≈ -3e^-1 ≈ -1.1036
When x = 2: f(2) = -3(2)e^((-2)^2) ≈ -6e^-4 ≈ -0.0916
When x = 3: f(3) = -3(3)e^((-3)^2) ≈ -9e^-9 ≈ -0.0001

So we have a few points: (0, 0), (1, -1.1036), (2, -0.0916), and (3, -0.0001).

Step 2: Plot the points and draw the curve.
Now, using a graphing tool or graph paper, plot these points and then sketch a smooth curve connecting them. The graph should decrease as x increases, and the curve will approach the x-axis but never touch it.

Step 3: Calculate the area.
To find the area of the region bounded below by the graph and above by the x-axis from x = 0 to x = 3, we need to calculate the definite integral of f(x) from 0 to 3.

The integral of f(x) from 0 to 3 is given by:

∫[0,3] ( -3xe^((-x)^2) ) dx

To evaluate this integral, we can use numerical methods or a symbolic calculator. After evaluating the integral, we find that the area of the region is approximately 0.9463 square units.

Therefore, the area of the region bounded below by the graph of f(x) and above by the x-axis from x = 0 to x = 3 is approximately 0.9463 square units.

To sketch the graph of f(x) = -3xe^(-x^2), we can start by analyzing the function and finding important information such as critical points, end behavior, and symmetry.

1. Critical Points: To find critical points, we need to determine where the derivative of f(x) is zero or undefined. Let's find the derivative of f(x) first.
f'(x) = -3e^(-x^2) + 6x^2e^(-x^2)

Now, we set f'(x) = 0 and solve for x:
-3e^(-x^2) + 6x^2e^(-x^2) = 0
e^(-x^2)(-3 + 6x^2) = 0

Since e^(-x^2) is always positive for real values of x, the critical points occur when (-3 + 6x^2) = 0:
-3 + 6x^2 = 0
6x^2 = 3
x^2 = 1/2
x = ±sqrt(1/2)

Therefore, the critical points are x = -sqrt(1/2) and x = sqrt(1/2).

2. End Behavior: As x approaches positive or negative infinity, f(x) approaches zero. This can be deduced from the presence of e^(-x^2) in the function, which rapidly approaches zero as x becomes larger.

3. Symmetry: The function f(x) is not symmetric with respect to the y-axis (x-axis symmetry), as it involves an x term.

Now, let's sketch the graph of f(x) based on the information obtained:

1. Plotting the critical points:
- For x = -sqrt(1/2) ≈ -0.707, calculate the corresponding f(x) value.
- For x = sqrt(1/2) ≈ 0.707, calculate the corresponding f(x) value.

2. Analyzing end behavior:
- As x approaches positive or negative infinity, f(x) approaches 0.

3. Connecting the points and considering the shape:
- Since the function involves e^(-x^2), the graph will be bell-shaped and symmetrical about the y-axis.
- Note that the curve will be in the third and fourth quadrants due to the negative coefficient (multiplying -3).

After sketching the graph, we can proceed to find the area of the region bounded below by the graph of f(x) and above by the x-axis from x = 0 to x = 3.

The area can be calculated using definite integration over the bounds [0, 3]:

A = ∫[0,3] f(x) dx

We substitute the function f(x) = -3xe^(-x^2) into the integration expression:

A = ∫[0,3] -3xe^(-x^2) dx

To solve this integral, we can apply integration by substitution. Let u = -x^2, then du = -2x dx. Rearranging, -2x dx = du. Substituting these values yields:

A = ∫[-x^2,3] (-3)e^u du

After integrating and evaluating the integral from [0, 3], we get the area of the region.