Find the error resulting from approximation by Trapezoidal Rule:

integral (from 0 to 1) sqrt(1+ x^3) dx
.... compute the results for n=8

8 trapezoids --> delta x = 1/8

x ______ y * f

0.0 1 * 1/2 ->.5
1/8 1.000976
1/4 1.00778
3/8 1.02603
1/2 1.06066
5/8 1.11541
3/4 1.19242
7/8 1.29225
8/8 1.41421*1/2 --> 0.707106

add them and multiply by 1/8

8.902633/8 =1.113

now do for rel
int (1+x^3)^.5 dx
= 1.11145
No, I used Wolfram, what a mess
see
http://www.wolframalpha.com/input/?i=integrate+%281%2Bx^3%29^.5+dx++from+x%3D0+to+1

error = 1.113 -1.111 = .002
or about 0.2 percent

Get it ?

Oh by the way, here is the recipe for the indefinite integral . put in

(1+x^3)^.5 dx
http://www.wolframalpha.com/input/?i=int+x^5+dx&lk=3

To find the error resulting from the approximation by the Trapezoidal Rule, we need to compare the actual value of the integral with the approximation obtained using the rule.

The Trapezoidal Rule states that the approximation of an integral is given by:

∫(from a to b) f(x) dx ≈ Δx/2 * [f(a) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(b)]

where Δx = (b - a) / n is the width of each subinterval, and n is the number of subintervals.

To compute the results for n = 8, we can follow these steps:

Step 1: Determine the width of each subinterval
Δx = (b - a) / n
Δx = (1 - 0) / 8
Δx = 1/8

Step 2: Compute the approximated value
Approximation ≈ Δx/2 * [f(0) + 2f(x1) + 2f(x2) + ... + 2f(x6) + f(1)]

We need to compute the values of f(x) at the interval's endpoints and the midpoints of each subinterval.

f(0) = sqrt(1 + 0^3) = sqrt(1) = 1

For the midpoints, we compute x values:
x1 = a + Δx/2 = 0 + (1/8)/2 = 1/16
x2 = a + Δx/2 = 0 + (1/8)/2 = 3/16
x3 = a + Δx/2 = 0 + (1/8)/2 = 5/16
x4 = a + Δx/2 = 0 + (1/8)/2 = 7/16

f(x1) = sqrt(1 + (1/16)^3)
f(x2) = sqrt(1 + (3/16)^3)
f(x3) = sqrt(1 + (5/16)^3)
f(x4) = sqrt(1 + (7/16)^3)

For the final endpoint:
f(1) = sqrt(1 + 1^3) = sqrt(2)

Now, we can plug these values into the formula:

Approximation ≈ (1/8)/2 * [1 + 2f(x1) + 2f(x2) + 2f(x3) + 2f(x4) + f(1)]

Compute the values of f(x1), f(x2), f(x3), f(x4) using the previously calculated values of x1, x2, x3, x4.

Finally, calculate the result for n = 8.

To find the error resulting from the approximation, you need to know the exact value of the integral and compare it with the approximated value obtained using the Trapezoidal Rule.