Evaluate the following integration: I(f) = integral sign from 0 to 20 of e^(-x) dx
1. Analytically
2. Rectangle method with h= 10,5,4,2,1.
3. Mid-point method with h= 10,5,4,2,1.
4. Trapezoidal method with h= 10,5,4,2,1.
5. Simpson's method with h= 10,5,4,2,1.
6. Using analytical and numerical solutions of 2) to 5) to calculate error e = | I(f) - I(f) | [absolute value of (analytical - numerical)]
Thanks a ton. =)
I will attempt #1.
1.
∫e-xdx
=-e-x+C
So for the limits 0 to 20, we get
I=[-e-x] 0 to 20
=-e-20-(-e-0)
=1-e-20
For the rest, use the appropriate method, and evaluate e-x at specified intervals to get the required answers.
If you have difficulty with any one of the methods, post your attempt to get further help.
1. Analytically:
To evaluate the definite integral analytically, we first need to find the antiderivative of the function f(x) = e^(-x).
The antiderivative of e^(-x) is -e^(-x).
Evaluating this antiderivative at the upper limit (20) and lower limit (0) of integration gives:
I(f) = [-e^(-x)] evaluated from 0 to 20
= -e^(-20) - (-e^(-0))
= -e^(-20) + 1
2. Rectangle method:
The rectangle method approximates the definite integral by dividing the interval [0, 20] into subintervals of equal width and approximating the function by a rectangle with height equal to the function value at a representative point within each subinterval.
With h = 10, the interval [0, 20] is divided into 2 subintervals.
The representative points within the subintervals are at x = 5 and x = 15.
Approximation using h = 10:
I(f) ≈ (f(5) + f(15)) * h
= (e^(-5) + e^(-15)) * 10
Similarly, you can calculate the approximations using h = 5, h = 4, h = 2, and h = 1.
3. Mid-point method:
The mid-point method is similar to the rectangle method, but it uses the value of the function at the midpoint of each subinterval to approximate the integral.
With h = 10, the interval [0, 20] is divided into 2 subintervals.
The midpoints within the subintervals are at x = 2.5 and x = 12.5.
Approximation using h = 10:
I(f) ≈ (f(2.5) + f(12.5)) * h
= (e^(-2.5) + e^(-12.5)) * 10
Similarly, you can calculate the approximations using h = 5, h = 4, h = 2, and h = 1.
4. Trapezoidal method:
The trapezoidal method approximates the definite integral by dividing the interval [0, 20] into subintervals and approximating the function by a trapezoid with the function values at the endpoints as the heights.
With h = 10, the interval [0, 20] is divided into 2 subintervals.
The endpoints within the subintervals are at x = 0, 10 and x = 10, 20 respectively.
Approximation using h = 10:
I(f) ≈ [(f(0) + f(10)) / 2 + (f(10) + f(20)) / 2] * h
= [(e^(-(0)) + e^(-10)) / 2 + (e^(-10) + e^(-20)) / 2] * 10
Similarly, you can calculate the approximations using h = 5, h = 4, h = 2, and h = 1.
5. Simpson's method:
Simpson's method approximates the definite integral using quadratic polynomials within each subinterval.
With h = 10, the interval [0, 20] is divided into 2 subintervals.
The endpoints and the midpoint within each subinterval are at x = 0, 10, 20 and x = 5, 15 respectively.
Approximation using h = 10:
I(f) ≈ (h / 3) * [(f(0) + 4 * f(5) + f(10)) + (f(10) + 4 * f(15) + f(20))]
= (10 / 3) * [(e^(-(0)) + 4 * e^(-5) + e^(-10)) + (e^(-10) + 4 * e^(-15) + e^(-20))]
Similarly, you can calculate the approximations using h = 5, h = 4, h = 2, and h = 1.
6. Error calculation:
To calculate the error, e, you can subtract the analytical solution obtained in step 1 from each of the numerical approximations obtained in steps 2-5, taking the absolute value:
e = | I(f) - I(f) |
Substitute the numerical and analytical values into this equation for each of the approximations obtained in steps 2-5 to calculate the corresponding errors.