The length of a curve for 0 <= x <= 20 is given by the integral from 0 to 20 of √(1 + 84x^4) dx. If this curve contains the point (1,11), what is the equation of the curve?

My answer is 3x^3 + 8.

Your answer looks good, if you meant

√(1 + 81x^4)

To find the equation of the curve, we need to first integrate the given expression:

∫ √(1 + 84x^4) dx

Let's break down the steps to calculate this integral.

1. Start by simplifying the expression inside the square root:

√(1 + 84x^4) = √(1^2 + (2√21x^2)^2) = √(1^2 + (2√21x^2)^2)

2. Now, substitute u = 2√21x^2:

du/dx = 2√21 * 2x
dx = du / (4√21 * x)

3. Rewrite the integral in terms of u:

∫ √(1 + (2√21x^2)^2) dx = ∫ √(1 + u^2) * (du / (4√21 * x))

4. Simplify the integral:

∫ √(1 + u^2) * (du / (4√21 * x)) = (1 / (4√21)) ∫ √(1 + u^2) * (1 / x) du

5. Now, integrate:

Let's denote the integral as I:

I = (1 / (4√21)) ∫ √(1 + u^2) * (1 / x) du

Using a trigonometric substitution, let u = tanθ, du = sec^2θ dθ:

I = (1 / (4√21)) ∫ √(1 + tan^2θ) * (1 / x) sec^2θ dθ

Using the trigonometric identity sec^2θ = 1 + tan^2θ, we simplify further:

I = (1 / (4√21)) ∫ √(1 + tan^2θ) * (1 / x) (1 + tan^2θ) dθ

I = (1 / (4√21)) ∫ √(1 + tan^2θ) * (1 + tan^2θ) / x dθ

I = (1 / (4√21x)) ∫ (1 + tan^2θ) dθ

I = (1 / (4√21x)) [θ + tanθ] + C

6. Now we can substitute back u = 2√21x^2:

I = (1 / (4√21x)) [θ + tanθ] + C

Since we are integrating with respect to θ, we need to express θ in terms of u:

u = 2√21x^2 = tanθ
θ = arctan(2√21x^2)

Substituting back in:

I = (1 / (4√21x)) [arctan(2√21x^2) + tan(arctan(2√21x^2))] + C

Using the identity tan(arctan(x)) = x, we simplify further:

I = (1 / (4√21x)) [arctan(2√21x^2) + 2√21x^2] + C

7. Finally, substitute the limits of integration (0 and 20) and solve for C:

I = (1 / (4√21x))[arctan(2√21x^2) + 2√21x^2] + C

When x = 20, we have:

I = (1 / (4√21 * 20))[arctan(2√21 * (20^2)) + 2√21 * 20^2] + C

Simplifying further:

I = (1 / (80√21))[arctan(2√21 * 400) + 1600√21] + C

Since the curve contains the point (1, 11), we can equate this point to the equation:

(1 / (80√21))[arctan(2√21 * 400) + 1600√21] + C = 11

Solving for C, we find:

C = 11 − (1 / (80√21))[arctan(2√21 * 400) + 1600√21]

Therefore, the equation of the curve is:

I = (1 / (4√21x))[arctan(2√21x^2) + 2√21x^2] + (11 − (1 / (80√21))[arctan(2√21 * 400) + 1600√21])