Can any show me step by step on how to get this? I keep on getting different answers... Thank You!
Use Simpson's rule with n=10 to estimate the arc length of y=x^(-1/3), for 1 <= x < 6.
Certainly! I can guide you step by step on how to use Simpson's rule to estimate the arc length of the function y=x^(-1/3), for 1 <= x < 6 with n=10.
Step 1: Define the function
The function we need to find the arc length for is y=x^(-1/3).
Step 2: Find the first and last x-values
In this case, our given range is 1 <= x < 6, so the first x-value is 1 and the last x-value is 6.
Step 3: Calculate the interval size (h)
The interval size (h) is the difference between the first and last x-values divided by the number of subintervals (n). In this case, n=10, so h = (6 - 1) / 10 = 0.5.
Step 4: Find the y-values for each x-value
Now we need to calculate the y-values corresponding to each x-value. For this step, you will need to calculate the value of y for each x-value using the formula y = x^(-1/3).
Here's a table to help you with the calculation:
```
| x | y |
|-------|--------|
| 1 | 1 |
| 1.5 | 0.874|
| 2 | 0.793|
| 2.5 | 0.741|
| 3 | 0.701|
| 3.5 | 0.668|
| 4 | 0.641|
| 4.5 | 0.617|
| 5 | 0.596|
| 5.5 | 0.577|
| 6 | 0.561|
```
Step 5: Apply Simpson's rule
Now, we can apply Simpson's rule to estimate the arc length.
The formula for Simpson's rule is:
L ≈ (h/3) * [y0 + 4y1 + 2y2 + 4y3 + ... + 2yn-2 + 4yn-1 + yn]
Since n=10, we have 11 x-values, so there are 10 subintervals.
Using the values from the table, we can substitute the y-values into the formula:
L ≈ (0.5/3) * [1 + 4*0.874 + 2*0.793 + 4*0.741 + 2*0.701 + 4*0.668 + 2*0.641 + 4*0.617 + 2*0.596 + 4*0.577 + 0.561]
Now, you can calculate the value inside the brackets and simplify the expression to find the estimated arc length.