Fun with Integration

Calculus Level 4

Solve your way through the following steps and submit the result of (c) as your answer.

(a) First, prove that for all values of $$n\in\mathbb{R}$$

$\int _{ 0 }^{ 1 }{ (n^{ 2 }x^{ n }+2nx^{ n }+x^{ n }-1) } dx\equiv n.$

The logical reasoning behind this proof will help in parts (b) and (c).

(b) Sketch, on the same axes, the curves representing $$f(x)=16x^3-1$$ and $$x=1.$$

(c) Express $$f(x)$$ in the form $$x^n (n+1)^2-1,$$ and hence, without integrating or making use of any approximations, state the area enclosed between the two curves and the coordinate axes.

Note: This problem can easily be worked out with a calculator, online maths website or by integrating directly. But that will take all the fun out of it! The question is based on the typical calculus A-level syllabus. Please follow the rules and arrive to the answer by following all the below steps chronologically.

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