Find the area of the region bounded by the curve y=x^2*e^x and the x-axis, 0≤x≤1
clearly, that's
∫[0,1] x^2 e^x dx
use integration by parts.
The only difficulty lies in integrating x^2 e^x
I used integration by parts twice and got
∫x^2 e^x dx = x^2 e^x - 2x e^x + 2e^x
= e^x( x^2 - 2x + 2)
so ∫ x^2 e^x dx from 0 to 1
= [e^x(x^2 - 2x + 2)] from 0 to 1
= e(1-2+2) - e^0(0-0+2)
= e - 2
To find the area of the region bounded by the curve y = x^2*e^x and the x-axis, we can use the definite integral.
The first step is to find the points of intersection between the curve and the x-axis. This is where y = 0. Setting y = 0, we have:
0 = x^2*e^x
This equation can be solved by factoring out x^2 and using the property that e^x is never equal to 0. We have two solutions:
x = 0 or e^x = 0
The second equation has no real solutions, so the only point of intersection is x = 0.
Next, we set up the definite integral to calculate the area. Since the curve is above the x-axis for the given interval, we have:
Area = ∫[0,1] (x^2*e^x) dx
To evaluate this integral, we can use integration techniques such as integration by parts or substitution. In this case, if we perform integration by parts, we can let u = x^2 and dv = e^x dx. Then, du = 2x dx and v = e^x.
Using the formula for integration by parts, we have:
∫(x^2*e^x) dx = x^2*e^x - ∫(2x*e^x) dx
We continue the integration by parts on the remaining integral until we have an expression that can be easily evaluated.
Integrating 2x*e^x, we can apply integration by parts again, with u = 2x and dv = e^x dx. This gives us:
∫(2x*e^x) dx = 2x*e^x - ∫(2*e^x) dx
Integrating 2*e^x gives us:
∫(2*e^x) dx = 2*e^x
Putting it all together, we have:
∫(x^2*e^x) dx = x^2*e^x - (2x*e^x - 2*e^x)
Expanding and simplifying, we get:
∫(x^2*e^x) dx = x^2*e^x - 2x*e^x + 2*e^x
Now, we can evaluate this expression at the upper and lower bounds of the definite integral:
Area = [x^2*e^x - 2x*e^x + 2*e^x] evaluated from 0 to 1
Area = (1^2*e^1 - 2*1*e^1 + 2*e^1) - (0^2*e^0 - 2*0*e^0 + 2*e^0)
Simplifying further, we have:
Area = (e - 2e + 2e) - (0 - 0 + 2)
Area = 2
Therefore, the area of the region bounded by the curve y = x^2*e^x and the x-axis, for the interval 0 ≤ x ≤ 1, is 2 square units.