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

Answer:

a) $\pm 8\frac{x^{5}}{5}+C$

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

Step-by-step 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

$[f'(x)]^{2} = 64x^{8}\\f'(x)=\pm 8x^{4}\\f(x)= \pm8\frac{x^{5}}{5}+C$

Part B

Compare (2) with standard form

$[f'(x)]^{2}=64cos^{2}(2x)\\f'(x)= \pm 8cos(2x)\\f(x)=\pm 8\frac{sin(2x)}{2}+C\\f(x)= \pm 4sin(2x)+C$