The length of the arc of the semi cubical parabola y2 = x3 from its vertex to the point (1, 1) i s ______ . (correct upto two decimal places)Correct answer is '1.44'. Can you explain this answer?

Question

Answer

Diff. the equation of the curve

So length of the arc of the cuive

Answer

Explanation:

The arc length of the curve can be found using the formula for arc length of a curve:

Arc Length = ∫ [sqrt(1 + (dy/dx)²)] dx

In this case, the curve is represented by the equation y² = x³.

1. Find dy/dx:
Differentiating y² = x³ with respect to x, we get:
2y(dy/dx) = 3x²
dy/dx = (3x²)/(2y)

2. Find the integral:
Substitute dy/dx into the arc length formula:
Arc Length = ∫ [sqrt(1 + (3x²/2y)²)] dx
Simplify the integrand:
= ∫ [sqrt(1 + (9x⁴)/(4y²))] dx
= ∫ [sqrt(1 + (9x⁴)/(4x³))] dx
= ∫ [sqrt(1 + (9x)/(4))] dx

3. Limits of integration:
To find the arc length from the vertex to the point (1, 1), we need to find the value of x when y = 1:
1² = x³
x = 1

4. Calculate the arc length:
Now, plug in the limits of integration:
Arc Length = ∫ [sqrt(1 + (9x)/(4))] dx, from x=0 to x=1
= ∫ [sqrt(1 + (9x)/(4))] dx, from 0 to 1
= ∫ [sqrt(1 + (9x)/(4))] dx, from 0 to 1
≈ 1.44 (rounded to two decimal places)

Therefore, the length of the arc of the semi-cubical parabola from its vertex to the point (1, 1) is approximately 1.44 units.

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