9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = 3 ln x− x²/24 on [1, 6]

Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)

b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy

Volumes without calculus Solve the following problems with and without calculus. A good picture helps.

b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?

Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).

b. How is this formula changed if x0>b?

Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = −8x−3 on [−2, 6] (Use calculus.)

Find the arc length of the line y = 4−3x on [−3, 2] using calculus and verify your answer using geometry.

Find the arc length of the line y = 2x+1 on [1, 5] using calculus and verify your answer using geometry.

3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.

y = 2 cos 3x on [−π,π]