How do you find the length of the curve for #y=x^2# for (0, 3)?

Answer 1

Arc Length #= 1/4sinh^-1 6 + 3/2sqrt(37)#

The Arc Length #l# is given by the integration formula # l=int_a^b sqrt(1+(dy/dx)^2)dx#
With #y=x^2 => dy/dx=2x#. And so:
# l = int_0^3 sqrt(1+(2x)^2)dx # # :. l = int_0^3 sqrt(1+4x^2)dx #

I will quote the result, but if you want to see how to perform the integration, please use this link

# l = [sinh^-1(2x)/4 + (xsqrt(4x^2+1))/2]_0^3 # # :. l = (sinh^-1 6/4 + (3sqrt(36+1))/2) - (sinh^-1 0/4 + 0)# # :. l = (sinh^-1 6/4 + (3sqrt(37))/2) - (0 + 0)# # :. l = 1/4sinh^-1 6 + 3/2sqrt(37)#
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Answer 2

To find the length of the curve (y = x^2) from (x = 0) to (x = 3), we use the arc length formula:

[L = \int_{x_1}^{x_2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx]

For (y = x^2), (\frac{dy}{dx} = 2x). Substituting into the formula and integrating from (x = 0) to (x = 3):

[L = \int_{0}^{3} \sqrt{1 + (2x)^2} , dx]

[L = \int_{0}^{3} \sqrt{1 + 4x^2} , dx]

This integral may be evaluated using trigonometric substitution. Let (x = \frac{1}{2}\tan(\theta)), then (dx = \frac{1}{2}\sec^2(\theta) , d\theta). Substituting:

[L = \int_{0}^{\arctan(6)} \sqrt{1 + \tan^2(\theta)} \cdot \frac{1}{2}\sec^2(\theta) , d\theta]

[L = \int_{0}^{\arctan(6)} \sqrt{\sec^2(\theta)} \cdot \frac{1}{2}\sec^2(\theta) , d\theta]

[L = \int_{0}^{\arctan(6)} \frac{1}{2}\sec^3(\theta) , d\theta]

[L = \frac{1}{2} \int_{0}^{\arctan(6)} \sec^3(\theta) , d\theta]

This integral can be evaluated using integration techniques such as integration by parts or tables of integrals. Once evaluated, you'll obtain the length of the curve.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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