Final answer:
The formula for a midpoint Riemann sum with 30 intervals for the given integral is M_n = δx [Σ(sin(x_1^*) + ... + sin(x_n^*))], where δx is the interval width and x_i^* are the midpoints of each of the 30 intervals.
Step-by-step explanation:
The question asks for the formula for a midpoint Riemann sum to estimate the integral of sin(x) over the interval from π to 4π with 30 intervals. The formula can be written as:
M_n = δx [Σ(sin(x_1^*) + sin(x_2^*) + ... + sin(x_n^*))]
where:
- δx is the width of each interval and can be calculated by δx = (Π - 4Π)/30
- ∃* represents the midpoint of each interval
- n indicates the number of intervals, which in this case is 30
In this specific case, the midpoint Riemann sum formula can be applied by first calculating δx, then determining the x_i^* values for each interval (which will be between π and 4π), and lastly, summing up the sine of these midpoints multiplied by δx.