Question

What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give all functions that satisfy the conditions integral a to b squareroot1 + 64x^8 dx integral a to b squareroot1 + 64 cos^2(2x)dx Choose the correct answer below. Select all that apply. A. Y = 8x^4 + C B. y = -8x^5/5 + C C. y = -8x^4 + C D. y = 8x^5/5 + C E. y = 32x^3 + C F. y = -32x^3 + C G. y = 4x^6/15 + C H. y = -4x^6/15 + C

Answer 1

a)
`\pm 8(x^(5))/(5)+C`

b)
`\pm 4 sin(2x) + C`

Explanation:

Given integrals are:

`a) \int\limits^a_b {\sqrt{{1+64x^(8)} } \, dx` ---(1)

`b) \int\limits^a_b {\sqrt{1+64cos^(2)(2x)} } \, dx`---- (2)

Standard form

`L= \int\limits^a_b {\sqrt{1+(f'(x))^(2)} } \, dx`

Part A

compare (1) with standard form

`[f'(x)]^(2) = 64x^(8)\\f'(x)=\pm 8x^(4)\\f(x)= \pm8(x^(5))/(5)+C`

Part B

Compare (2) with standard form

`[f'(x)]^(2)=64cos^(2)(2x)\\f'(x)= \pm 8cos(2x)\\f(x)=\pm 8(sin(2x))/(2)+C\\f(x)= \pm 4sin(2x)+C`