Question

Why did I get this question wrong?

Answer 1

You set up was almost accurate. Remember the arc length formula:

If f'(y) is continuous on the interval [a,b], then the length of the curve x = f(y), a ≤ y ≤ b should be;

L = ∫ᵇ ₐ √1 + [f'(y)]^2 * dy

We have to find the length of the curve given x = √y - 2y, and 1 ≤ y ≤ 4. You can tell your limits would be 1 to 4, and you are right on that part. But f'(y) would be rather...

f'(y) = 1/(2√y) - 2

So the integral would be:

∫⁴₁ √1 + (1/(2√y) - 2)² dy

Using a calculator we would receive the solution 5.832. Their is a definite curve, as represented below;

Answer 2

Answer:

s = ∫₁⁴ √(1 + (1/(2√y) − 2)²) dy

s = 5.8319

Step-by-step explanation:

Arc length is:

s = ∫ ds

s = ∫ √(1 + (dx/dy)²) dy

Find dx/dy:

x = √y − 2y

dx/dy = 1/(2√y) − 2

Substitute:

s = ∫₁⁴ √(1 + (1/(2√y) − 2)²) dy

Evaluate with a calculator:

s = 5.8319