To solve the problem of finding the area between the curve (f(x) = x^3 - 3x^2 + 2x) and the x-axis from (x=0) to (x=2), follow these steps:

Step 1: Factor the function to find roots

(f(x) = x(x^2 - 3x + 2) = x(x-1)(x-2)).
Roots are at (x=0,1,2).

• Between (0) and (1): (f(x) > 0) (positive).
• Between (1) and (2): (f(x) < 0) (negative).

Step 2: Compute the integral of (f(x))

[ \int (x^3 - 3x^2 + 2x)dx = \frac{x^4}{4} - x^3 + x^2 + C ]

Step 3: Calculate the area

Area = (\int_0^1 f(x)dx + \left|\int_1^2 f(x)dx\right|)

• (\int_0^1 f(x)dx = \left(\frac{1}{4} - 1 + 1\right) - 0 = \frac{1}{4})
• (\int_1^2 f(x)dx = \left(\frac{16}{4} - 8 + 4\right) - \left(\frac{1}{4} -1 +1\right) = -\frac{1}{4})
• Absolute value: (\left|-\frac{1}{4}\right| = \frac{1}{4})

Total Area

(\frac{1}{4} + \frac{1}{4} = \frac{1}{2})

Answer: (\boxed{\dfrac{1}{2}})

作者声明:本文包含人工智能生成内容。