Find the arclength of [tex]y=3 x^{3 / 2}[/tex] on [tex]1 \leq x \leq 2[/tex]
Answer (1)
Find the derivative of y = 3 x 3/2 , which is y ′ = 2 9 x .
Square the derivative: ( y ′ ) 2 = 4 81 x .
Calculate 1 + ( y ′ ) 2 = 1 + 4 81 x .
Integrate ∫ 1 2 1 + 4 81 x d x to find the arc length, which is approximately 5.57653954636410 .
Explanation
Problem Setup We are asked to find the arc length of the curve y = 3 x 3/2 on the interval 1 ≤ x ≤ 2 . The formula for the arc length of a curve y = f ( x ) from x = a to x = b is given by L = ∫ a b 1 + ( f ′ ( x ) ) 2 d x where f ′ ( x ) is the derivative of f ( x ) with respect to x .
Finding the Derivative First, we need to find the derivative of y = 3 x 3/2 with respect to x .
d x d y = 3 ⋅ 2 3 x 2 3 − 1 = 2 9 x 2 1 So, y ′ = 2 9 x .
Squaring the Derivative Next, we need to find ( y ′ ) 2 .
( y ′ ) 2 = ( 2 9 x ) 2 = 4 81 x
Adding 1 to the Square of the Derivative Now, we need to find 1 + ( y ′ ) 2 .
1 + ( y ′ ) 2 = 1 + 4 81 x
Taking the Square Root We need to find the square root of 1 + ( y ′ ) 2 .
1 + ( y ′ ) 2 = 1 + 4 81 x
Evaluating the Integral Now, we need to integrate 1 + 4 81 x with respect to x from 1 to 2 .
L = ∫ 1 2 1 + 4 81 x d x Let u = 1 + 4 81 x , so d x d u = 4 81 , and d x = 81 4 d u . When x = 1 , u = 1 + 4 81 = 4 85 . When x = 2 , u = 1 + 4 81 ( 2 ) = 1 + 2 81 = 2 83 .
L = ∫ 4 85 2 83 u ⋅ 81 4 d u = 81 4 ∫ 4 85 2 83 u 2 1 d u = 81 4 [ 3 2 u 2 3 ] 4 85 2 83 = 243 8 [ u 2 3 ] 4 85 2 83 L = 243 8 [ ( 2 83 ) 2 3 − ( 4 85 ) 2 3 ] = 243 8 [ 2 2 83 166 − 4 4 85 85 ] = 243 8 [ 2 2 83 83 − 8 85 85 ] L = 243 8 [ ( 2 83 ) 3/2 − ( 4 85 ) 3/2 ] ≈ 5.57653954636410
Final Answer The arc length of the curve y = 3 x 3/2 on the interval 1 ≤ x ≤ 2 is approximately 5.57653954636410 .
Examples
Imagine you are designing a roller coaster track. You have a section of the track that follows the curve y = 3 x 3/2 between the points x = 1 and x = 2 . To estimate the amount of material needed for this section, you need to calculate the arc length of the curve. This calculation helps ensure you have enough material to build the track safely and efficiently. Understanding arc length is crucial in engineering for estimating lengths of curved structures.
