Respuesta :
Ok, here we go. Pay attention. The formula for the arc length is [tex]AL= \int\limits^b_a { \sqrt{1+( \frac{dy}{dx})^2 } } \, dx [/tex]. That means that to use that formula we have to find the derivative of the function and square it. Our function is y = 4x-5, so y'=4. Our formula now, filled in accordingly, is [tex]AL= \int\limits^2_1 { \sqrt{1+4^2} } \, dx [/tex] (that 1 is supposed to be negative; not sure if it is til I post the final answer). After the simplification we have the integral from -1 to 2 of [tex] \sqrt{17} [/tex]. Integrating that we have [tex]AL= \sqrt{17}x [/tex] from -1 to 2. [tex]2 \sqrt{17}-(-1 \sqrt{17} ) [/tex] gives us [tex]3 \sqrt{17} [/tex]. Now we need to do the distance formula with this. But we need 2 coordinates for that. Our bounds are x=-1 and x=2. We will fill those x values in to the function and solve for y. When x = -1, y=4(-1)-5 and y = -9. So the point is (-1, -9). Doing the same with x = 2, y=4(2)-5 and y = 3. So the point is (2, 3). Use those in the distance formula accordingly: [tex]d= \sqrt{(2-(-1))^2+(3-(-9))^2} [/tex] which simplifies to [tex]d= \sqrt{9+144}= \sqrt{153} [/tex]. The square root of 153 can be simplified into the square root of 9*17. Pulling out the perfect square of 9 as a 3 leaves us with [tex]3 \sqrt{17} [/tex]. And there you go!
The arc length of the curve is [tex]3\sqrt{17}[/tex].
Given
The length of the curve y = 4x − 5, −1 ≤ x ≤ 2.
Arc length;
A part of a curve or a part of a circumference of a circle is called Arc. All of them have a curve in their shape.
The given graph y = 4x-5 is the straight line.
The straight line distance between the points is;
[tex]\rm Distance \ point = (-2, f(-2))= (-2, 4(-2)-5)=(-2, \ (-8-5))= (-2, \ -13)\\\\And (1, \ f(1))= (1, \ 4(1)-5)= (1, \ 4-5)= (1, -1)[/tex]
Therefore,
The distance between these points is given by;
[tex]))^2\rm Arc \ length =\sqrt{[1-(-2)]^2+[-1-(-13)^2]} \\\\Arc \ length=\sqrt{(1+2)^2+(-1+13)^2}\\\\Arc \ length=\sqrt{(3)^2+(12)^2}\\\\ Arc \ length=\sqrt{9+144}\\\\ Arc \ length=\sqrt{153}\\\\Arc \ length=3\sqrt{17}[/tex]
Hence, the arc length of the curve is [tex]3\sqrt{17}[/tex].
To know more about arc length click the link given below.
https://brainly.com/question/24251184
