f(1)= 20, f(3)=13, f(5)=15, f(7)=16, f(9)=11, on [0,6]
a, used midpint rule with n=5 to estimate intergral form 0 to 10 f(x)dx
b, use trapezoidal rule with n=4 to estimate intergral from 1 to 9 f(x)dx
c, used simpson's rule with n=4 to estimate intergal from 1 to 9 (x)dx
To estimate the integral using different numerical methods, we need to calculate the values of f(x) at specific points within the given interval. Since the given function values are provided at odd integer points in the interval [0,6], we will interpolate the function values at even integer points as well.
Given function values:
f(1) = 20
f(3) = 13
f(5) = 15
f(7) = 16
f(9) = 11
Interpolated function values:
f(0) = (20 + 13)/2 = 16.5
f(2) = (13 + 15)/2 = 14
f(4) = (15 + 16)/2 = 15.5
f(6) = (16 + 11)/2 = 13.5
f(8) = (11 + f(7))/2 = (11 + 16)/2 = 13.5
a) Midpoint Rule with n = 5 to estimate integral from 0 to 10 f(x)dx:
Using the Midpoint Rule, the formula to estimate the integral is given by:
∫[a,b] f(x)dx ≈ (b-a)/n * ∑[i=0 to n-1] f((x_i+1 + x_i)/2)
In this case, a = 0, b = 10, and n = 5
∆x = (10 - 0)/5 = 2
The estimated integral using the Midpoint Rule is:
∫[0,10] f(x)dx ≈ 2 * [f(1) + f(3) + f(5) + f(7) + f(9)]
= 2 * [20 + 13 + 15 + 16 + 11]
= 2 * 75
= 150
b) Trapezoidal Rule with n = 4 to estimate the integral from 1 to 9 f(x)dx:
Using the Trapezoidal Rule, the formula to estimate the integral is given by:
∫[a,b] f(x)dx ≈ ∆x/2 * [f(x_0) + 2 * ∑[i=1 to n-1] f(x_i) + f(x_n)]
In this case, a = 1, b = 9, and n = 4
∆x = (9 - 1)/4 = 2
The estimated integral using the Trapezoidal Rule is:
∫[1,9] f(x)dx ≈ 2/2 * [f(1) + 2 * (f(3) + f(5) + f(7)) + f(9)]
= [20 + 2 * (13 + 15 + 16) + 11]
= [20 + 2 * 44 + 11]
= [20 + 88 + 11]
= 119
c) Simpson's Rule with n = 4 to estimate the integral from 1 to 9 f(x)dx:
Using Simpson's Rule, the formula to estimate the integral is given by:
∫[a,b] f(x)dx ≈ ∆x/3 * [f(x_0) + 2 * ∑[i=1 to n/2] f(x_2i-1) + 4 * ∑[i=1 to n/2-1] f(x_2i) + f(x_n)]
In this case, a = 1, b = 9, and n = 4
∆x = (9 - 1)/4 = 2
The estimated integral using Simpson's Rule is:
∫[1,9] f(x)dx ≈ 2/3 * [f(1) + 2 * (f(3) + f(5)) + 4 * f(7) + f(9)]
= [2/3 * (20 + 2 * (13 + 15) + 4 * 16 + 11)]
= [2/3 * (20 + 2 * 28 + 64 + 11)]
= [2/3 * (20 + 56 + 64 + 11)]
= [2/3 * 151]
= 100.67 (rounded to two decimal places)
Therefore, the estimates for the given integrals are:
a) ∫[0,10] f(x)dx ≈ 150
b) ∫[1,9] f(x)dx ≈ 119
c) ∫[1,9] f(x)dx ≈ 100.67