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Calculus 1 : Length of Line

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Example Question #1 : Length Of Line

Consider a function with first-derivative:

$\frac{dy}{dx}=\tan{x}$ .

Which integral can calculate the length along this curve between x=0 and $x=2\pi$ ?

Possible Answers:

$L=\int_{0}^{2\pi}{\frac{1}{\cos{x}}}\quad dx$

$L=\int_{0}^{2\pi}{\tan^2{x}}\quad dx$

$L=\int_{0}^{2\pi}{\frac{1}{\sin{x}}}\quad dx$

$L=\int_{0}^{2\pi}{\sqrt{1+\sin^2}}\quad dx$

This is not possible with the information given.

Correct answer:

$L=\int_{0}^{2\pi}{\frac{1}{\cos{x}}}\quad dx$

Explanation:

To determine the length of a curve between two points, we evaluate the integral

$L = \int_0^{2\pi}{\sqrt{1+\left(\frac{dy}{dx} \right )^2}}dx$

There are many reasons this works, but we'll give an informal explanation here: Canvas2

If we divide this curve into three line-segments, we can see that they become more and more similar to the original curve. By adding up all the little hypotenuses, we can arrive at the length of the curve. Note, that

$\sqrt{dx^2 + dy^2} = \sqrt{1+\left(\frac{dy}{dx} \right )^2}dx$

If we think of the integration symbol as a sum of infinitely small parts, this gets us the formula for length:

$L = \int_a^b{\sqrt{1+\left(\frac{dy}{dx} \right )^2}}dx$ .

Returning to the problem, we plug the derivative into the length formula:

$L=\int_{0}^{2\pi}{\sqrt{1+\tan^2{x}}}\quad dx$

substitute:

$L=\int_{0}^{2\pi}{\sqrt{\left( \frac{\cos{x}}{\cos{x}} \right)^2+\left(\frac{\sin{x}}{\cos{x}} \right )^2}}\quad dx$

Simplify:

$L=\int_{0}^{2\pi}{\sqrt{\frac{\cos^2{x}}{\cos^2{x}}+\frac{\sin^2{x}}{\cos^2{x}}}}\quad dx$

$L=\int_{0}^{2\pi}{\sqrt{\frac{\cos^2{x}+\sin^2{x}}{\cos^2{x}}}}\quad dx$

By trigonometric identities we get:

$L=\int_{0}^{2\pi}{\sqrt{\frac{1}{\cos^2{x}}}}\quad dx$

$L=\int_{0}^{2\pi}{\frac{1}{\cos{x}}}\quad dx$

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Example Question #2 : Length Of Line

Find the length of the line segment between points A and B:

Possible Answers:

$3\sqrt{2}$

$\sqrt{58}$

$\sqrt{338}$

Correct answer:

$\sqrt{338}$

Explanation:

The distance between two points can be easily found using the Distance Formula:

$D=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

Applying the points we are given to this formula results in:

$D=\sqrt{((-5)-2)^2+(10-(-7))^2}=\sqrt{(-7)^2+(17)^2}=\sqrt{338}$

This is one of the answer choices.

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Calculus 1 : Length of Line

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #1 : Length Of Line

Example Question #2 : Length Of Line

Find the length of the line segment between points A and B:

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