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.

The answer is 3.

1 solution

Vincent Moroney
Jun 27, 2018

$\begin{aligned} I(n) = & \int_0^1 ( x^n(n+1)^2 -1) \,dx \\ = & \frac{1}{n+1} (n+1)^2 -1 \\ = & n \end{aligned}$ Now we want to use this information to evaluate $\displaystyle \int_0^1 (16x^3 - 1)\,dx$ . First notice that $16 = (3+1)^2$ so what we have is $\displaystyle \int_0^1 (x^3(3+1)^2 -1) \,dx$ , meaning $n=3 \, \, \, \, \therefore \, \, \, \, \text{I(3)} = \boxed{3}$ .

Fun with Integration

The answer is 3.

1 solution

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