Final answer:
The integration problem can be solved analytically by finding the antiderivative, numerically using the trapezoidal rule with four divisions, and also by applying Simpson's 1/3 rule. Since the specific function f(x) and numerical values for a and b are not provided, the general methods for these calculations are described instead of actual numerical results.
Step-by-step explanation:
The student's question involves calculating the definite integral of a function f(x) from a given lower limit a to an upper limit b. Here, the function and the limits are not clearly provided in the question, so we cannot compute an actual integral. However, we can describe the general method to accomplish the task for an arbitrary function. The analytical solution involves finding the antiderivative of the function and evaluating it at the limits of integration. To apply the trapezoidal rule, one would divide the interval [a, b] into n equal parts and then calculate the area of the trapezoids formed under the curve, summarizing their areas to approximate the integral. Simpson's 1/3 rule is another numerical method that uses parabolic arcs instead of straight lines to approximate the area under the curve and typically provides a more accurate estimate than the trapezoidal rule, especially when used with a greater number of intervals.