Final answer:
To approximate the definite integral, we can use three numerical integration methods: Trapezoidal Rule, Midpoint Rule, and Simpson's Rule.
Step-by-step explanation:
The student's question is related to determining the work done by a variable force applied along a displacement with the aid of numerical integration methods. The force function is given as F(x) = (10 N)sin[(0.1 m⁻¹)x], and we're looking to find the work done from x = 0 to x = 10 m.
Since this requires integration and we're not given an antiderivative, we resort to numerical methods like the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule, all with n = 10 subdivisions. To perform these techniques, one would start by splitting the interval [0, 10] into 10 equal parts, calculate the function values at endpoints and midpoints as necessary, and then apply the formulas associated with each rule to approximate the integral, thus estimating the work done by the force.
Note: To provide a complete numerical answer, the specific values of the function at the required points would need to be calculated and inserted into the formulas for each of these rules.