Length of a curve - Revision history

Milllo at 00:12, 2 April 2021

2021-04-02T00:12:34Z

2021-04-02T00:12:16Z

2021-04-02T00:11:57Z

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 length of curve = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx   (this equation can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here])
+-------
 For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup>

← Older revision		Revision as of 00:12, 2 April 2021
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	length of curve = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx (this equation can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here])		length of curve = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx (this equation can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here])
		+

	For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup>		For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup>

← Older revision		Revision as of 00:11, 2 April 2021
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−	The length of a curve from point a to point b can be found using an integral of the first derivative of the equation:	+	The length of a curve from point a to point b can be found using an integral of the first derivative of the equation: equation of curve = f(x), first derivative = f '(x)
−
−	equation of curve = f(x)
−
−	first derivative = f '(x)

	length of curve = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx (this equation can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here])		length of curve = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx (this equation can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here])

@@ Line 5: / Line 5: @@
 first derivative = f '(x)
-length of curve = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx   (can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here]
+length of curve = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx   (this equation can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here])
 For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup>

@@ Line 5: / Line 5: @@
 first derivative = f '(x)
-length of curve = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx
+length of curve = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx   (can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here]
 For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup>

@@ Line 1: / Line 1: @@
 The length of a curve from point a to point b can be found using an integral of the first derivative of the equation:
-equation  = f(x)
+equation of curve  = f(x)
 first derivative = f '(x)
-length = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx
+length of curve = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx
 For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup>

← Older revision		Revision as of 00:07, 2 April 2021
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	<font size = "+2"><span>∫</span></font>sqr(1+(30e<sup>−0.3t</sup>)<sup>2</sup>)dt		<font size = "+2"><span>∫</span></font>sqr(1+(30e<sup>−0.3t</sup>)<sup>2</sup>)dt
		+
		+	entering this equation into the integral calculator [https://www.integral-calculator.com/ here] gives 77.8 degrees, which is about the same value calculated in the previous example using integration of the rate of change!

← Older revision		Revision as of 00:05, 2 April 2021
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	so the length of the curve from 0 to 5 minutes would be:		so the length of the curve from 0 to 5 minutes would be:

−	<font size = "+2"><span>∫</span></font>sqr(1+30e<sup>−0.3t</sup><sup>2</sup>)dt	+	<font size = "+2"><span>∫</span></font>sqr(1+(30e<sup>−0.3t</sup>)<sup>2</sup>)dt

← Older revision		Revision as of 23:39, 1 April 2021
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	length = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx		length = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx
		+
		+	For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup>
		+
		+	so the length of the curve from 0 to 5 minutes would be:
		+
		+	<font size = "+2"><span>∫</span></font>sqr(1+30e<sup>−0.3t</sup><sup>2</sup>)dt