math calc2

Arc Length

Theorem 1 Let f be continuously differentiable on the closed interval [a,b]. Then the arc length of the curve y = f(x) over [a,b] is given by

L =

∫ b a

√ 1 + [f′(x)]2 dx

1. Let C be the line segment defined by the equation y = 2x + 1 over −1 ≤ x ≤ 3.

a) Use Theorem 1 to calculate the arc length of C.

b) Use the ordinary distance formula to find the arc length of C.

2. Compute the arc length of the curve y = 1 3 (2 + x2)

3 2 over the interval 0 ≤ x ≤ 3.

� 3–5 These problems require the use of integral tables. In each instance, cite the table form that you use.

3. Find the arc length of y = ln x over [1, √

3].

4. Prove that the circumference C of a circle with radius r is given by C = 2πr.

5. A hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation

y = 180 − x2

45

until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express you answer correct to the nearest tenth of a meter.

Hint: Your integral will require a substitution to bring it into a table form.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. Use Simpson’s Rule with n = 6 to estimate the length of y = sin x over [0,π] to 5 decimal places.

Hint: To streamline the process of entering Simpson’s formula into your calculator, enter your arc- length square-root function as Y1. Then on your home screen, you can call and evaluate the function by entering VARS, right-arrow for Y-VARS, ENTER, ENTER. This puts Y1 on your home screen which you may now evaluate using parenthesis. For example, Y1(5).

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Solutions to Selected Problems

1. a) 4 √

5

b) 4 √

5

2. 12

3. ln

(√ 3 + √

6

3

) + 2 −

√ 2 ≈ 0.92

4. Proof

5. ≈ 209.1

6. ≈ 3.81940

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