integrate x^(1/2)e^x.dx from 0 to 4, n=12
1) the trapezoidal rule
You will need to calculate f(x) at thirteen points: x = 0, 1/3, 2/3, 1, 1 1/3, ... 3 2/3 and 4. The width of each trapezoid is h = 1/3.
Add up the values of f(x) at each point except the endpoints, add
[f(0) + f(4)]/2 to that, and multiply by h.
To approximate the integral using the trapezoidal rule, follow these steps:
1. Divide the interval [0, 4] into n equal subintervals. Since n = 12 in this case, each subinterval will have a width of (4 - 0) / 12 = 1/3.
2. Evaluate the function at the endpoints of each subinterval. In this case, evaluate the function x^(1/2)e^x at x = 0, 1/3, 2/3, 1, 4/3, etc., up to x = 4.
3. Use the formula for the trapezoidal rule to compute the approximate integral. The formula is:
I ≈ (h/2) * [f(x0) + 2*f(x1) + 2*f(x2) + ... + 2*f(xn-1) + f(xn)]
Where h is the width of each subinterval, f(x) is the function you are integrating, and x0, x1, x2, ..., xn are the endpoints of the subintervals.
4. Apply the formula using the values obtained from step 2. The first term and the last term should be divided by 2.
I ≈ (1/6) * [f(0) + 2*f(1/3) + 2*f(2/3) + 2*f(1) + 2*f(4/3) + ... + 2*f(4) + f(4)]
Calculate the corresponding values of f(x) for each endpoint and plug them into the formula.
5. Simplify and calculate the approximate value of the integral.
Note: Since the formula for the trapezoidal rule involves adding up the values of the function at multiple points, it can be time-consuming to do by hand. Using a calculator or a computer program that performs numerical integration can make it quicker and more accurate.