Approximate the arc length of the curve y=(1/4)x^4 over the interval [1,2] using the Trapezoidal Rule T8.
I was never taught how to use the Trapezoidal Rule, so I was surprised to find this problem in my most recent Calculus assignment. How would you solve a problem like this?
Thanks in advance to anyone who's willing to help out.
This is a WebWork problem for my Calculus 2 class and all the answers I got from online calculators weren't considered correct, which is why I'm asking for some help on here.
To approximate the arc length of a curve using the Trapezoidal Rule, you need to follow these steps:
Step 1: Determine the interval of integration. In this case, the interval is given as [1,2].
Step 2: Find the derivative of the function. Taking the derivative of y = (1/4)x^4, we get dy/dx = x^3.
Step 3: Calculate the integrand. The integrand for arc length is given by sqrt(1 + (dy/dx)^2). In this case, it becomes sqrt(1 + (x^3)^2).
Step 4: Approximate the arc length using the Trapezoidal Rule. The Trapezoidal Rule for approximating arc length is given by the formula:
L ≈ h/2 * [f(x0) + 2*f(x1) + 2*f(x2) + ... + 2*f(xn-1) + f(xn)],
where h = (b-a)/n is the width of each trapezoid, n is the number of trapezoids, and f(x) is the integrand.
In this case, with T8, n = 8.
Step 5: Calculate the values of x and f(x). Evaluate f(x) = sqrt(1 + (x^3)^2) for x = 1, 1.125, 1.25, ..., 1.875, 2.
Step 6: Use the Trapezoidal Rule formula to find the approximation. Plug in the calculated values of f(x) into the Trapezoidal Rule formula and evaluate.
That's it! Follow these steps to approximate the arc length of the curve y = (1/4)x^4 over the interval [1,2] using the Trapezoidal Rule T8.
To approximate the arc length of a curve over an interval using the Trapezoidal Rule, you first need to divide the interval into smaller subintervals and then calculate the length of each subinterval. By summing up the lengths of these subintervals, you can obtain an estimate of the total arc length.
To use the Trapezoidal Rule specifically, follow these steps:
1. Divide the interval [a, b] into equal subintervals. In this case, the interval is [1, 2]. Since you will be using T8, you will divide it into 8 subintervals, making each subinterval of length Δx = (2-1)/8 = 1/8.
2. Determine the coordinates of the points on the curve for each subinterval. In this case, the given curve is y = (1/4)x^4. Plug in the x-values corresponding to each subinterval to find the corresponding y-values.
3. Calculate the length of each subinterval using the formula for the arc length of a curve segment, which is given by √(Δx^2 + Δy^2), where Δx is the change in x and Δy is the change in y between two adjacent points on the curve.
4. Add up the lengths of all the subintervals to obtain an approximation of the total arc length.
Now let's go through the calculations for this specific problem using the Trapezoidal Rule T8:
Subinterval 1: x = 1/8
y = (1/4)(1/8)^4 = 1/2048
Δx = 1/8
Δy = (1/4)(1/8)^4 - 0 = 1/2048
Subinterval 2: x = 2/8
y = (1/4)(2/8)^4 = 1/256
Δx = 1/8
Δy = (1/4)(2/8)^4 - (1/4)(1/8)^4 = 1/256 - 1/2048 = 7/32768
Continue this process for the remaining subintervals.
Once you have the Δx and Δy for each subinterval, you can calculate the length of each subinterval using the formula √(Δx^2 + Δy^2). Add up these lengths to get an estimate for the arc length over the interval.
Finally, it's important to note that the Trapezoidal Rule is an approximation method, and as you increase the number of subintervals, the estimate usually becomes more accurate. In this case, since you are using T8, the estimate will be better than if you were using T4 or T2.
I hope this explanation helps you understand how to use the Trapezoidal Rule to approximate the arc length of a curve. If you have any further questions, feel free to ask!