To estimate the definite integral using the same rule for n=30, we can start by calculating the interval width h for n=10 using the given rule, and then use the same value of h to estimate the integral with n=30.
A) Left Rule:
The Left Rule divides the interval into n subintervals of equal width and evaluates the function at the left endpoint of each subinterval.
The interval width for n=10 using the Left Rule can be calculated as:
h = (b - a) / n = (4 - 0) / 10 = 0.4
To estimate the integral with n=30 using the Left Rule, we can use the same interval width:
h = 0.4
The number of subintervals for n=30 can be calculated as:
n = (b - a) / h = (4 - 0) / 0.4 = 10
Therefore, for n=30 using the Left Rule, we will have 10 subintervals.
B) Trapezoid Rule:
The Trapezoid Rule divides the interval into n subintervals of equal width and approximates the area under the curve as the sum of the areas of trapezoids formed by the function values at the endpoints of each subinterval.
The interval width for n=10 using the Trapezoid Rule can be calculated as:
h = (b - a) / n = (4 - 0) / 10 = 0.4
To estimate the integral with n=30 using the Trapezoid Rule, we can use the same interval width:
h = 0.4
The number of subintervals for n=30 can be calculated as:
n = (b - a) / h = (4 - 0) / 0.4 = 10
Therefore, for n=30 using the Trapezoid Rule, we will have 10 subintervals.
In both cases, we use the same width h because the same approximation rule is used. The only difference is that the number of subintervals, n, changes to a higher value (30).
Keep in mind that as the number of subintervals increases, the accuracy of the approximation usually improves, but this depends on the behavior of the function being integrated.