Find the length of the curve over the interval .Round your answer to two decimal places.

Question

Find the length of the curve  over the interval  .Round your answer to two decimal places. ​

Quizplus · Accepted Answer

To find the length of the curve, we use the formula:
$$L = \int_{a}^{b} \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$$
Plugging in the given values, we have:
$$L = \int_{11eac44f_ebed_c49c_bedf_2fa2d5b14567_TB7497_11}^{11eac44f_ebed_9d8b_bedf_e750a603ed54_TB7497_11} \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$$
We can find $\frac{dy}{dx}$ by differentiating $y$ with respect to $x$, which gives us:
$$\frac{dy}{dx} = \frac{1}{2}x(12-x^2)^{-1/2}$$
Plugging this back into the formula, we get:
$$L = \int_{11eac44f_ebed_c49c_bedf_2fa2d5b14567_TB7497_11}^{11eac44f_ebed_9d8b_bedf_e750a603ed54_TB7497_11} \sqrt{1+\frac{1}{4x^2(12-x^2)}}dx$$
This integral can be solved using a substitution, but it is quite difficult to do so. Instead, we can use numerical integration. Using a calculator or computer program, we get:
$$L \approx 19.05$$
Therefore, the answer is (B).

Find the Length of the Curve Over the Interval