Use Simpson’s rule to estimate ∫_0^2▒〖1/8 e〗^(x^2 ) dx with a maximum error of 0.1
To use Simpson's rule to estimate the definite integral with a maximum error of 0.1, we need to divide the interval [0, 2] into equal subintervals and apply the formula for Simpson's rule.
Simpson's rule is based on approximating the function with quadratic polynomials over each subinterval. It uses the average of the function values at the endpoints and the midpoint of each subinterval to estimate the integral.
The formula for Simpson's rule is given by:
∫_a^b▒f(x) dx ≈ h/3 [f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + ... + 2f(b-h) + 4f(b-h) + f(b)]
Here, h represents the step size, which is determined by dividing the interval into equal subintervals. In this case, we want to find the step size that satisfies the maximum error of 0.1.
To find the step size, we can use the error formula for Simpson's rule:
E = -(b-a) * h^4 * f''(c) / 180
where E is the maximum error, f''(c) is the second derivative of the function, and c is some value between a and b.
In this case, let's find the second derivative of the function f(x) = (1/8)e^(x^2). We differentiate f(x) twice:
f''(x) = d^2/dx^2 [(1/8)e^(x^2)]
= (1/8)(2x)e^(x^2) + (1/8)e^(x^2) * d/dx(2x)
= (1/4)xe^(x^2) + (1/4)e^(x^2)
The second derivative is f''(x) = (1/4)xe^(x^2) + (1/4)e^(x^2).
Now, we substitute the values into the error formula:
0.1 = -(2-0) * h^4 * [(1/4)xe^(x^2) + (1/4)e^(x^2)] / 180
Simplifying the equation, we have:
0.1 = -2h^4 * [(1/4)xe^(x^2) + (1/4)e^(x^2)] / 180
Next, we solve for h by isolating it in the equation:
0.1 = -h^4 * [(1/4)xe^(x^2) + (1/4)e^(x^2)] / 90
Re-arranging the equation, we get:
h^4 = 0.1 * 90 / [(1/4)xe^(x^2) + (1/4)e^(x^2)]
Simplifying further:
h^4 = 4.5 / [(1/4)xe^(x^2) + (1/4)e^(x^2)]
h^4 = 18 / [xe^(x^2) + e^(x^2)]
To find the maximum error of 0.1, we need h^4 to be equal to 18 / [xe^(x^2) + e^(x^2)]. We can use a numerical method like the bisection method or iterate over various values of h until we find a value that satisfies the equation.
Once we have determined the value of h, we can proceed to apply Simpson's rule formula to estimate the integral:
∫_0^2▒[1/8 e^(x^2)] dx ≈ h/3 [f(0) + 4f(h) + 2f(2h) + ... + 2f(2-h) + 4f(2-h) + f(2)]
Evaluate the function at each point using f(x) = (1/8)e^(x^2), and then sum up the weighted function values to get the estimated value of the integral.
Please note that finding the exact step size might not be straightforward, as the equation is transcendental. Hence, it might be necessary to employ numerical methods to approximate the optimal step size.