The first subinterval is of width 2 (sec); the 2nd's width is 3 sec; the 3rd's width is 4 sec; the last subinterval's width is 1.iRaid said:
The Trapezoidal Rule is a numerical method used to approximate the value of a definite integral. It involves dividing the area under a curve into trapezoids and summing their areas to estimate the total area.
To use the Trapezoidal Rule, you first need to determine the limits of integration and the function that you want to approximate the integral of. Then, divide the interval into smaller subintervals and calculate the area of each trapezoid using the formula A = (1/2)(b1 + b2)h, where b1 and b2 are the lengths of the two parallel sides and h is the height. Finally, sum the areas of all the trapezoids to get an approximation of the integral.
The error associated with the Trapezoidal Rule is dependent on the number of subintervals used in the approximation. The more subintervals, the smaller the error. The error can be calculated using the formula E = -(b-a)^3/12n^2*f''(c), where (b-a) is the interval, n is the number of subintervals, and f''(c) is the second derivative of the function at some point c within the interval.
Yes, the Trapezoidal Rule can be used for any continuous function. However, it is most accurate for functions that are smooth and have no sharp changes in direction.
The accuracy of the Trapezoidal Rule depends on the number of subintervals used. With a large number of subintervals, it can be more accurate than other numerical methods such as the Midpoint Rule or Simpson's Rule. However, it is generally less accurate than more complex methods such as Gaussian Quadrature.