×
Back to all chapters

Definite Integrals

The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.

Definite Integrals: Level 3 Challenges

$\displaystyle \int_{0}^{20} \Bigl(\lfloor x \rfloor \{x\} \Bigr) \ dx = \ ?$

Details and assumptions:

• Every $$x\in \mathbb{R}$$ can be written as $$x=\lfloor x \rfloor + \{x\}$$.

• $$\lfloor x \rfloor$$ denotes greatest integer less than or equal to $$x$$.

• $$\{x\}$$ is the fractional part of $$x$$.

Let $$I_m = \displaystyle \int_0^{2\pi} \cos(x) \cos(2x) \dots \cos(mx) dx$$. What is the sum of all integers $$100 \leq m \leq 110$$ such that $$I_m \neq 0$$?

$\displaystyle{ S }_{ N }=\sum _{k=1}^{N}\frac1k$

Let $$S_N$$ satisfy the equation above. What is the value of $$\displaystyle \lim _{ n\rightarrow \infty }{ \left( { S }_{ 2n }-{ S }_{ n } \right) }$$?

$$f(x) = \begin{cases}{1-|x|}, && {|x|\>\le\>1} \\ {|x|-1,} && {|x|>1}\end{cases}$$

$$g(x) = f(x-1)+f(x+1)$$.

Given the two functions above, what is the value of $$\displaystyle \int _{ -3 }^{ 5 }{ g(x) \mathrm{d}x }$$?

$\large \int_0^{\pi /2} ( \sin^{2014}x - \cos^{2014}x ) \,dx = \, ?$

×