ength
a. Set up an integral for the length of the curve.
b. Graph the curve to see what it looks like.
c. Use a graphing utility or computer to find the length of the curve numerically.
from to
a.

To find the length of the curve given by the equation , we need to first express in terms of . Rewriting the equation yields . This is a quadratic equation in , which we can solve using the quadratic formula . Here, .

Once we have expressed as functions of , we can find the derivative . The length of the curve from point to point can be set up as:

Now, don’t forget to graph the curve after setting up the integral! It helps to visualize the section you’re calculating the length for, and reinforces the understanding of the function’s behavior.

As for the numerical calculation of the length, using a graphing utility or computer algebra system can significantly simplify the process. Devices like Desmos or software like Mathematica will allow you to input the integral you derived and compute the length of the curve directly, which will give you an accurate result without the tedium of manual calculations. Happy graphing!