Question

Find the length of the curve yequalsthree fourths x Superscript 4 divided by 3 Baseline minus three eighths x Superscript 2 divided by 3 Baseline plus 9 for 1less than or equalsxless than or equals8.

Answer 1

Length of the curve = 1.207 units

Explanation:

y =
`(3)/(4) x^(4/3) - (3)/(8) x^(2/3) + 9` for 1 ≤x≤8

The formula for finding the length of a curve is:

Length of a curve =
`\int\limits^a_b {\sqrt{1 + ((dy)/(dx))^(2) } } \, dx`

To compute the length we need (dy/dx)². So,

dy/dx =
`(4)/(3) * (3)/(4) x^((4/3) -1) - (2)/(3)*(3)/(8)x^((2/3) - 1)`

dy/dx =
`x^(1/3) - (1)/(4) x^(-1/3)`

(dy/dx)² = (
`x^(1/3) - (1)/(4) x^(-1/3)`)²

(dy/dx)² =
`x^(2/3) - (1)/(16) x^(-2/3)`

Length of a curve =
`\int\limits^a_b {\sqrt{1 + ((dy)/(dx))^(2) } } \, dx`

=
`\int\limits^8_1 {\sqrt{1 + x^(2/3) - (1)/(16) x^(-2/3) } } \, dx`

=
`\int\limits^8_1`[(1 + x^(2/3) - (1/16)(x^-2/3)]^1/2 dx

= 2/3 [(1 + x^(2/3) - (1/16)(x^-2/3)]^3/2 * [(2/3)*x^-1/3 + (1/24)*x^(-5/3)]

Applying limits 1 to 8:

(2/3)*[(1 + (8)^(2/3) - (1/16)(8)^(-2/3)]^3/2 * [(2/3)*(8)^(-1/3) + (1/24)*(8)^(-5/3)] - (2/3)*[(1 + (1)^(2/3) - (1/16)(1)^(-2/3)]^3/2 * [(2/3)*(1)^(-1/3) + (1/24)*(1)^(-5/3)]

= (2/3)*(11.12)*(0.3346) - (2/3)*(2.6968)*(0.7083)

= 2.4805 - 1.2734

Length of the curve = 1.207 units