A theoretical physicist is studying the relativistic correction factor, C(v)C(v)C(v), for a particle's effective mass at different velocities vvv. The factor is modeled by the function C(v)=116+vC(v) = \frac{1}{\sqrt{16 + v}}C(v)=16+v1.
Find the first three terms, in ascending powers of vvv, of the binomial expansion of C(v)C(v)C(v).
Hence, find the first three terms of the binomial expansion of C(−v2)=116−v2C(-v^2) = \frac{1}{\sqrt{16 - v^2}}C(−v2)=16−v21.
Using your answer to part (b), find an approximation for ∫02116−v2 dv\int_{0}^{2} \frac{1}{\sqrt{16 - v^2}} \, dv∫0216−v21dv, giving your answer to seven decimal places.
(i) A research assistant decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used. Explain clearly whether this new approximation will be an overestimate, an underestimate, or if it is impossible to tell.
(ii) The assistant goes on to use the expansion from part (b) to find an approximation for ∫05116−v2 dv\int_{0}^{5} \frac{1}{\sqrt{16 - v^2}} \, dv∫0516−v21dv. Explain why this calculation is invalid.