Internal 1/sqrt(1+x^3) from [0,2] and n=10

there are lots of online calculators which you can use to check your results.

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.