The approximation to a definite integral using n=10 is 2.346; the exact value is 4.0. If the approximation was found using each of the following rules, use the same rule to estimate the integral with n=30.
A) Left Rule
B) Trapezoid Rule
The section deals with approximation errors of definite integrals.
The error in the first approximation is 1.654. I'm not sure if that is relevant in this problem, but I imagine it would be since that's the topic of the section.
If the original estimate was done by LEFT then the error is inversely proportional to the number of steps and the n = 30 error is (10/30) * -1.654 = -.551, approximately. So the estimate for n = 30 would be -.551 + 4.000 = 3.449
If the original estimate was done by TRAP then the error is inversely proportional to the square of the number of steps and the n = 30 error is (10/30)^2 * -1.654 = -.184, approximately. So the estimate for n = 30 would be -.184 + 4.000 = 3.816
If the original estimate was done by SIMP then the error is inversely proportional to the fourth power of the number of steps and the n = 30 error is (10/30)^4 * -1.654 = -.020, approximately. So the estimate for n = 30 would be -.02 + 4.000 = 3.980
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.
To estimate the integral using the same rule with n=30, we can use the concept of proportionality between the number of subintervals (n) and the size of each subinterval (Δx).
A) Left Rule:
The Left Rule approximates the integral by using the left endpoint of each subinterval. The formula for the Left Rule is:
∫[a,b] f(x) dx ≈ Δx * [f(a) + f(a+Δx) + f(a+2Δx) + ... + f(b-Δx)]
Given that n=10 and the approximate value is 2.346, we can find the value of Δx:
Approximation = Δx * [f(a) + f(a+Δx) + f(a+2Δx) + ... + f(b-Δx)]
2.346 = Δx * [f(a) + f(a+Δx) + f(a+2Δx) + ... + f(b-Δx)]
Since we have 10 intervals, n=10, the size of each subinterval is:
Δx = (b - a) / n = (b - a) / 10
To estimate the integral with n=30 using the Left Rule, we need to find the new approximation value:
Δx_new = (b - a) / n_new = (b - a) / 30
Approximation_new = Δx_new * [f(a) + f(a+Δx_new) + f(a+2Δx_new) + ... + f(b-Δx_new)]
Substituting the values, we can calculate the new approximation:
Approximation_new = ((b - a) / 30) * [f(a) + f(a+((b - a) / 30)) + f(a+2((b - a) / 30)) + ... + f(b-((b - a) / 30))]
B) Trapezoid Rule:
The Trapezoid Rule approximates the integral by averaging the values at the endpoints of each subinterval and multiplying by the width of each subinterval. The formula for the Trapezoid Rule is:
∫[a,b] f(x) dx ≈ Δx * [(f(a) + f(a+Δx)) / 2 + (f(a+Δx) + f(a+2Δx)) / 2 + ... + (f(b-Δx) + f(b)) / 2]
Using the same logic as above, we find the new approximation with n=30:
Approximation_new = ((b - a) / 30) * [(f(a) + f(a+((b - a) / 30))) / 2 + (f(a+((b - a) / 30)) + f(a+2((b - a) / 30))) / 2 + ... + (f(b-((b - a) / 30)) + f(b)) / 2]
Note: The error in the approximation is related to the size of the subintervals (Δx), so it may be relevant to compare the error between the different rules. The error can be calculated using the error formulas for each rule.