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AP Calculus AB: 10.10.2 Finding Arc Lengths of Curves Given by Functions

Mathematics12 CardsCreated 3 months ago

This content focuses on computing the arc length of smooth curves defined by functions over specific intervals. It explains how to apply the arc length formula, highlights the role of derivatives and definite integrals, and provides strategies for simplifying complex integrals commonly encountered in arc length problems.

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Finding Arc Lengths of Curves Given by Functions

The arc length of a smooth curve given by the function f (x) between a and b is

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Key Terms

Term
Definition

Finding Arc Lengths of Curves Given by Functions

The arc length of a smooth curve given by the function f (x) between a and b is

note

  • Given this curve, compute the arc length between x = 1 and x = 2.

  • To find the arc length, you will need the arc length formu...

Set up the integral for the length of the smooth arc y=tanx on [0, 1].

∫10√1+sec4xdx.

Find the length of the smooth arc y = (4 − x^2/3)^3/2 on [1, 4].

6 ^3√2−3

Set up the integral for the length of the smooth arc y = e x on [0, 10].

∫10 0√1+e2xdx

Set up the integral for the length of the smooth arc y=x⋅sinx on [0, 1].

∫1 0√1+(sinx+x⋅cosx)2dx

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AP Calculus AB: 10.10.2 Finding Arc Lengths of Curves Given by Functions
TermDefinition

Finding Arc Lengths of Curves Given by Functions

The arc length of a smooth curve given by the function f (x) between a and b is

note

  • Given this curve, compute the arc length between x = 1 and x = 2.

  • To find the arc length, you will need the arc length formula. Notice that the arc length formula requires you to take the derivative, so do this first.

  • Once you have the derivative, put it into the formula with the correct limits of integration. The limits of integration are just the starting and ending x-values of the particular arc you want to measure.

  • Now you have the integral and are ready to find the actual arc length.

  • Notice that in this example the integral is kind of difficult to evaluate. Integrals involving arc length tend to be more complicated than other integrals because of the radical sign in the arc length formula. It’s a good idea to look for a way to eliminate the radical sign.

  • Here, the radical sign is cancelled by expressing the
    denominator and numerator in terms of a square.

  • Start by breaking the fraction into two pieces. The integral is easier to evaluate that way.

  • Now the integral can be evaluated using the power rule.

  • Plugging in the limits of integration and simplifying give you the arc length.

Set up the integral for the length of the smooth arc y=tanx on [0, 1].

∫10√1+sec4xdx.

Find the length of the smooth arc y = (4 − x^2/3)^3/2 on [1, 4].

6 ^3√2−3

Set up the integral for the length of the smooth arc y = e x on [0, 10].

∫10 0√1+e2xdx

Set up the integral for the length of the smooth arc y=x⋅sinx on [0, 1].

∫1 0√1+(sinx+x⋅cosx)2dx

Suppose that y1=f(x) and y2=f(x)+10are two smooth curves, where f is the same function in both y1 and y2. Let L1 and L2 be the arc lengths of y1 and y2, respectively on the interval [a, b]. Whichof the following is the correct relation ofL1 to L2?

L1=L2

Find the length of the curve described by y=2x^3/2 /3 on [0, 8].

52/3

Find the length of the smooth arc y=2(x2+1)^3/2 / 3 on [0, 2].

22/3

Compute the length of the smooth arc y=ln|cosx| on [0, π/4].

ln(√2+1)

Set up the integral for the length of the smooth arc y = x ^4 + 5 on [0, 4].

∫40√1+16x6dx

Set up the integral for the length of the smooth arcy=lnx on [2, 6].

∫62⎷1+1/x^2dx