10.
Explanation
The integral ∫[0 to 1] x * 1/(1 + x^2) dx using Simpson's rule with three and six points, we'll first divide the interval [0, 1] into subintervals and then apply Simpson's rule to approximate the integral. Simpson's rule is an approximation method for numerical integration.
The formula for Simpson's rule for a function f(x) on the interval [a, b] with n subintervals (n must be even) is:
3h[f(a)+4∑i=1n/2f(a+(2i−1)h)+2∑i=1n/2−1f(a+2ih)+f(b)]
where h is the width of each subinterval,
h=(b−a)/n.
Let's first use three points (n = 2) and then six points (n = 4) to approximate the integral:
Three Points (n = 2):
-
a=0
-
b=1
-
2n=2
-
ℎ=(1−0)/2=0.5h=(1−0)/2=0.5
≈0.53[0+4(0.5⋅1/(1+(0.5)2))+1⋅1/(1+(0.5)2)+1]I3≈30.5[0+4(0.5⋅1/(1+(0.5)2))+1⋅1/(1+(0.5)2)+1]
Simplify and calculate:
3≈0.53[0+4(0.5/(1.25))+1/(1.25)+1]≈0.7672I3≈30.5[0+4(0.5/(1.25))+1/(1.25)+1]≈0.7672
Six Points (n = 4):
-
a=0
-
b=1
-
4n=4
-
ℎ=(1−0)/4=0.25h=(1−0)/4=0.25
6≈0.253[0+4(0.25⋅1/(1+(0.25)2))+2(0.25⋅1/(1+(0.5)2))+4(0.25⋅1/(1+(0.75)2))+1⋅1/(1+(1)2)]I6≈30.25[0+4(0.25⋅1/(1+(0.25)2))+2(0.25⋅1/(1+(0.5)2))+4(0.25⋅1/(1+(0.75)2))+1⋅1/(1+(1)2)]
Simplify and calculate:
0.253[0+4(0.25/(1.0625))+2(0.25/(1.25))+4(0.25/(1.5625))+1/(2)]≈0.7833I6≈30.25[0+4(0.25/(1.0625))+2(0.25/(1.25))+4(0.25/(1.5625))+1/(2)]≈0.7833
So, using three points, the approximate value of the integral is 0.7672, and using six points, it is 0.7833.