Internal 1/sqrt(1+x^3) from [0,2] and n=10
(a) Use the Trapezoidal Rule to approximate the given integral with the specified value of n.
(b) Use the Midpoint Rule to approximate the given integral with the specified value of n.
(c) Use Simpson's Rule to approximate the given integral with the specified value of n.
To approximate the integral using different methods, we need to divide the interval [0, 2] into subintervals of equal width. Since n = 10, we will divide the interval into 10 subintervals of width Δx = (2-0)/10 = 0.2.
(a) Trapezoidal Rule:
The formula for approximating an integral using the Trapezoidal Rule is:
∫[a,b] f(x) dx ≈ (Δx/2) [f(a) + 2f(x1) + 2f(x2) + ... + 2f(x(n-1)) + f(b)]
where x1, x2, ..., x(n-1) are the midpoints of the subintervals.
In this case, a = 0, b = 2, Δx = 0.2.
Plugging these values into the formula, we get:
∫[0,2] 1/√(1+x^3) dx ≈ (0.2/2) [f(0) + 2f(0.2) + 2f(0.4) + ... + 2f(1.8) + f(2)]
To evaluate this, we need to calculate the values of f(x) at each point. Let's calculate f(x) = 1/√(1+x^3) for each of the x values:
f(0) = 1/√(1+0^3) = 1
f(0.2) = 1/√(1+0.2^3) = 0.9759
f(0.4) = 1/√(1+0.4^3) = 0.9565
...
f(1.8) = 1/√(1+1.8^3) = 0.7400
f(2) = 1/√(1+2^3) = 0.7093
Now, we can substitute these values into the Trapezoidal Rule formula:
∫[0,2] 1/√(1+x^3) dx ≈ (0.2/2) [1 + 2(0.9759) + 2(0.9565) + ... + 2(0.7400) + 0.7093]
Simplifying this expression will give us the approximate value of the integral.
(b) Midpoint Rule:
The formula for approximating an integral using the Midpoint Rule is:
∫[a,b] f(x) dx ≈ Δx [f(x1) + f(x2) + ... + f(xn)]
where x1, x2, ..., xn are the midpoints of the subintervals.
In this case, a = 0, b = 2, Δx = 0.2.
Plugging these values into the formula, we get:
∫[0,2] 1/√(1+x^3) dx ≈ 0.2 [f(0.1) + f(0.3) + f(0.5) + ... + f(1.9)]
To evaluate this, we need to calculate the values of f(x) at each midpoint x.
(c) Simpson's Rule:
The formula for approximating an integral using Simpson's Rule is:
∫[a,b] f(x) dx ≈ (Δx/3) [f(a) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 2f(x(n-2)) + 4f(x(n-1)) + f(b)]
where x1, x2, ..., x(n-1) are the midpoints of the subintervals.
In this case, a = 0, b = 2, Δx = 0.2.
Plugging these values into the formula, we get:
∫[0,2] 1/√(1+x^3) dx ≈ (0.2/3) [f(0) + 4f(0.2) + 2f(0.4) + 4f(0.6) + ... + 4f(1.8) + 2f(2)]
To evaluate this, we need to calculate the values of f(x) at each x value.
After evaluating the respective formulas, you will obtain approximations of the given integral using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with n = 10.