Question

Find the exact length of the curve. y = 5 + 8x3/2, 0 ≤ x ≤ 1

Answer 1

The exact length of the curve
`\( y = 5 + 8x^(3/2) \)` over the interval
`\( 0 \leq x \leq 1 \)` is given by:
`L = (-1 + 145√(145))/(216)`
It is approximately
`\( 8.08 \)` when rounded to two decimal places.

Let's proceed with the solution to find the exact length of the curve
`\( y = 5 + 8x^(3/2) \)` over the interval
`\( 0 \leq x \leq 1 \)`.

Step 1: Formula for the Length of a Curve

The length
`\( L \)` of a curve given by
`\( y = f(x) \)` from
`\( x = a \) to \( x = b \)` is calculated using the formula:

`$$L = \int_(a)^(b) \sqrt{1 + \left((dy)/(dx)\right)^2} \, dx$$`

Step 2: Differentiating
`\( y \)`

First, we need to find the derivative
`\( (dy)/(dx) \) of \( y = 5 + 8x^(3/2) \)`.

`(dy)/(dx) = (d)/(dx)(5 + 8x^(3/2))`

Step 3: Integrating to Find the Length

After finding
`\( (dy)/(dx) \)`, we will substitute it into the formula for
`\( L \)` and evaluate the integral from
`\( x = 0 \) to \( x = 1 \)`.

Let's start by calculating the derivative
`\( (dy)/(dx) \)`.

The derivative
`\( (dy)/(dx) \)` of
`\( y = 5 + 8x^(3/2) \) is \( 12√(x) \)`.

Step 4: Setting Up the Integral

Now, we substitute this derivative into the formula for the length of the curve:

`$$L = \int_(0)^(1) \sqrt{1 + (12√(x))^2} \, dx$$`

This simplifies to:

`$$L = \int_(0)^(1) √(1 + 144x) \, dx$$`

Step 5: Evaluating the Integral

Let's evaluate this integral to find the exact length of the curve.

The exact length of the curve
`\( y = 5 + 8x^(3/2) \)` over the interval
`\( 0 \leq x \leq 1 \)` is given by:

`L = (-1 + 145√(145))/(216)`

Detailed Evaluation of the Integral

We need to evaluate the integral:

`L = \int_(0)^(1) √(1 + 144x) \, dx`

This integral can be solved by making a substitution that simplifies the square root term. Let's use the substitution
`\( u = 1 + 144x \)`. Then,
`\( du = 144 dx \)` or
`\( dx = (du)/(144) \)`. When
`\( x = 0 \), \( u = 1 \)`, and when
`\( x = 1 \), \( u = 145 \)`.

Substituting these into the integral, we get:

`L = \int_(1)^(145) √(u) \cdot (du)/(144)`

This simplifies to:

`L = (1)/(144) \int_(1)^(145) u^(1/2) \, du`

The integral of
`\( u^(1/2) \)` with respect to
`\( u \)` is
`\( (2)/(3) u^(3/2) \)`. Applying this to our integral, we get:

`L = (1)/(144) \left[ (2)/(3) u^(3/2) \right]_(1)^(145)`

Evaluating this from 1 to 145, we have:

`L = (1)/(144) \left[ (2)/(3) (145^(3/2)) - (2)/(3) (1^(3/2)) \right]`

Now, let's calculate this value to get the length of the curve up to two decimal places.

The exact length of the curve
`\( y = 5 + 8x^(3/2) \)` over the interval
`\( 0 \leq x \leq 1 \)` is approximately
`\( 8.08 \)` when rounded to two decimal places.

Answer 2

The exact length of the curve. y = 5 + 8x³/₂, 0 ≤ x ≤ 1 is (145³/₂ - 1)/216

To find the exact length of the curve. y = 5 + 8x³/₂, 0 ≤ x ≤ 1, we proceed as follows.

The exact length of a curve s = ∫√[1 + (dy/dx)²]dx.

So, since y = 5 + 8x³/₂,

dy/dx = d(5 + 8x³/₂)/dx

= d5/dx + d8x³/₂/dx

= 0 + 8 × 3/2x¹/₂

= 4 × 3x¹/₂

= 12x¹/₂

So, substituting this into s, we have

s = ∫√[1 + (dy/dx)²]dx.

s = ∫√[1 + (12x¹/₂)²]dx.

s = ∫₀¹√[1 + 144x]dx.

Now let u = 1 + 144x

du/dx = 144

dx = du/144

Also, when x = 0, u = 1 + 144(0) = 1 + 0 = 1

when x = 1, u = 1 + 144(1) = 1 + 144 = 145

So, substiting this into s, we have that

s = ∫₀¹√[1 + 144x]dx.

s = ∫₁¹⁴⁵√udu/144

s = [u(¹/₂ + 1)/(¹/₂ + 1)144]₁¹⁴⁵

s = [u³/₂/(3/2)144]₁¹⁴⁵

s = [2u³/₂/(3 × 144)]₁¹⁴⁵

s = [u³/₂/(3 × 72)]₁¹⁴⁵

s = [u³/₂/216]₁¹⁴⁵

s = (145³/₂ - 1³/₂)/216

s = (145³/₂ - 1)/216

So, the exact length is (145³/₂ - 1)/216