How do you find the arc length of the curve #y = 2x - 3#, #-2 ≤ x ≤ 1#?

Answer 1

The arc length is #3sqrt{5}\approx 6.7082#

Since the graph of #y=f(x)=2x-3# is a straight line, there's actually no need to use calculus. Instead, just find the straight-line distance between the points #(-2,f(-2))=(-2,-7)# and #(1,f(1))=(1,-1)#. The answer, by the distance formula (Pythagorean theorem), is
#sqrt{(-2-1)^2+(-7-(-1))^2}=sqrt{9+36}=sqrt{45}#
#=sqrt{9}sqrt{5}=3sqrt{5}\approx 6.7082#
If you want to confirm this with calculus, evaluate the integral #int_{-2}^{1}sqrt{1+(f'(x))^2}\ dx=int_{-2}^{1}sqrt{1+4}\ dx#
#=sqrt{5}x|_{-2}^{1}=sqrt{5}(1-(-2))=3sqrt{5}#
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Answer 2

To find the arc length of the curve ( y = 2x - 3 ) over the interval (-2 \leq x \leq 1), you can use the formula for arc length:

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

Where ( a ) and ( b ) are the limits of integration, and ( \frac{dy}{dx} ) is the derivative of ( y ) with respect to ( x ).

First, find ( \frac{dy}{dx} ):

[ \frac{dy}{dx} = 2 ]

Now, substitute into the arc length formula:

[ L = \int_{-2}^{1} \sqrt{1 + 2^2} , dx ] [ L = \int_{-2}^{1} \sqrt{1 + 4} , dx ] [ L = \int_{-2}^{1} \sqrt{5} , dx ]

Integrate:

[ L = \sqrt{5} \int_{-2}^{1} dx ] [ L = \sqrt{5} \left[x\right]_{-2}^{1} ] [ L = \sqrt{5} \left(1 - (-2)\right) ] [ L = \sqrt{5} \times 3 ] [ L = 3\sqrt{5} ]

So, the arc length of the curve over the given interval is ( 3\sqrt{5} ).

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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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