Final answer:
To approximate the integral of 7e^(1/x) from 1 to 2, one must divide the interval into 10 equal parts and apply the Trapezoidal and Midpoint Rule formulas, which require calculating the function value at specific points
Step-by-step explanation:
The question is asking to find the approximations using the Trapezoidal Rule (T10) and the Midpoint Rule (M10) for the integral from 1 to 2 of the function 7e^(1/x) dx. To use these rules, we need to divide the interval [1,2] into 10 equal parts and then apply the appropriate formulas for each method.
For the Trapezoidal Rule, the formula is:
Tn = (Delta x / 2) * (f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn))
Where Delta x is the width of each subinterval, x0, x1, ..., xn are the endpoints of the subintervals, and f(x) is our function.
For the Midpoint Rule, the formula is:
Mn = Delta x * (f((x0 + x1)/2) + f((x1 + x2)/2) + ... + f((xn-1 + xn)/2))
Once these calculations are performed, they will provide an approximation of the integral's value using 10 intervals for both the trapezoidal and midpoint rules.