Definite Integrals of Piecewise Functions

Definite Integrals of Piecewise Functions

Edited By Komal Miglani | Updated on Oct 15, 2024 12:58 PM IST

Integration is one of the important parts of Calculus, which applies to measuring the change in the function at a certain point. Mathematically, it forms a powerful tool by which slopes of functions are determined, the maximum and minimum of functions found, and problems on motion, growth, and decay, to name a few. These integration concepts have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.

Definite Integrals of Piecewise Functions
Definite Integrals of Piecewise Functions

Piecewise Definite integration

Definite integration calculates the area under a curve between two specific points on the x-axis.

Let f be a function of x defined on the closed interval [a, b]. F be another function such that $\frac{d}{d x}(F(x))=f(x)$ for all x in the domain of f, then $\int_a^b f(x) d x=[F(x)+c]_a^b=F(b)-F(a)$is called the definite integral of the function f(x) over the interval [a, b], where a is called the lower limit of the integral and b is called the upper limit of the integral.

Definite integrals have properties that relate to the limits of integration.

Property 1

If the upper and lower limits of integration are the same, the integral is just a line and contains no area, hence the value is 0. $\int_a^a f(x) d x=0$

Property 2

The value of the definite integral of a function over any particular interval depends on the function and the interval but not on the variable of the integration. $\int_a^b f(x) d x=\int_a^b f(t) d t=\int_a^b f(y) d y$

Property 3

If the limits of definite integral are interchanged, then its value changes by a minus sign only.

Property 4 (King's Property)

This is one of the most important properties of definite integration. $\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$

Property 5 (Piecewise Definite integration)

$\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\int_{\mathrm{a}}^{\mathrm{c}} \mathrm{f}(\mathrm{x}) \mathrm{dx}+\int_{\mathrm{c}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{dx}$ where $c \in \mathbb{R}$

This property is useful when the function is in the form of piecewise or discontinuous or non-differentiable at x = c in (a, b).
$
\begin{aligned}
& \text { Let } \quad \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{F}(\mathrm{x}))=\mathrm{f}(\mathrm{x}) \\
& \therefore \quad \int_a^c f(x) d x+\int_c^b f(x) d x \\
& =\left.\mathrm{F}(\mathrm{x})\right|_{\mathrm{a}} ^{\mathrm{c}}+\left.\mathrm{F}(\mathrm{x})\right|_{\mathrm{c}} ^{\mathrm{b}} \\
& =\mathrm{F}(\mathrm{c})-\mathrm{F}(\mathrm{a})+\mathrm{F}(\mathrm{b})-\mathrm{F}(\mathrm{c}) \\
& =\mathrm{F}(\mathrm{b})-\mathrm{F}(\mathrm{a}) \\
& =\int_a^b f(x) d x
\end{aligned}
$

The above property can be also generalized into the following form

$\int_a^b f(x) d x=\int_a^{c_1} f(x) d x+\int_{c_1}^{c_2} f(x) d x+\ldots+\int_{c_n}^b f(x) d x$

Property 6

$\int_0^a f(x) d x=\int_0^{a / 2} f(x) d x+\int_0^{a / 2} f(a-x) d x$

Proof:

From the previous property,

$
\int_0^a f(x) d x=\int_0^{a / 2} f(x) d x+\int_{a / 2}^a f(x) d x
$

Put $x=a-t \Rightarrow d x=-d t$ in the second integral, when $x=a / 2$, then $t=a / 2$ and when $x=a$, then $t=0$

$
\therefore \quad \begin{aligned}
\int_0^a f(x) d x & =\int_0^{a / 2} f(x) d x+\int_{a / 2}^0 f(a-t)(-d t) \\
& =\int_0^{\mathrm{a} / 2} \mathrm{f}(\mathrm{x}) \mathrm{dx}+\int_0^{\mathrm{a} / 2} \mathrm{f}(\mathrm{a}-\mathrm{t}) \mathrm{dt} \\
\int_0^a f(x) d x & =\int_0^{a / 2} f(x) d x+\int_0^{a / 2} f(a-x) d x
\end{aligned}
$

Recommended Video Based on Piecewise Definite Integration


Solved Examples Based on Piecewise Definite Integration:

Example 1: The integral $\int_0^\pi \sqrt{1+4 \sin ^2 \frac{x}{2}-4 \sin \frac{x}{2}} d x$ equals:

1) $4 \sqrt{3}-4$
2) $4 \sqrt{3}-4-\frac{\pi}{3}$
3) $\pi-4$
4) $\frac{2 \pi}{3}-4-4 \sqrt{3}$

Solution

As learnt in concept

Fundamental Properties of Definite integration -

If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.

$\int_b^a f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x$

- wherein

$\begin{aligned} & =\int_0^\pi \sqrt{\left(1-2 \sin \frac{x}{2}\right)^2}=\int_0^\pi\left|1-2 \sin \frac{x}{2}\right| d x \\ & \int_0^{\frac{\pi}{3}} \sqrt{\left(1-2 \sin \frac{x}{2}\right)}=\int_{\frac{\pi}{4}}^\pi\left(1-2 \sin \frac{x}{2}\right) d x \\ & \left(x+4 \cos \frac{x}{2}\right)_0^{\frac{\pi}{3}}-\left(x+4 \cos \frac{x}{2}\right)^\pi \\ & =4 \sqrt{3}-4-\frac{\pi}{3}\end{aligned}$

Example 2: The value of \int_{1}^{a}[x]f'(x)dx,\; a> 1,where\; [x] denotes the greatest integer not exceeding x is

1) af(a)- \left \{f(1)+f(2)+...+f([a]) \right \}\;

2) \; \; [a]f(a)-\left \{f(1)+f(2)+...+f([a]) \right \}

3) [a]f([a])-\left \{f(1)+f(2)+...+f(a) \right \}

4) af([a])-\left \{f(1)+f(2)+...+f(a) \right \}

Solution

As learnt in concept

Fundamental Properties of Definite integration -

If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.

$\int_b^a f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x$

- wherein


\left [ x \right ] has to be split into integral limits.

$\int_1^a[x] f^{\prime}(x) d x$

=$\int_1^2 f^{\prime}(x) d x+\int_2^3 2 f^{\prime}(x) d x+\ldots-\ldots+\int_{[a]}^a[a] f^{\prime}(x) d x$

$\begin{aligned} & f(2)-f(1)+2 f(3)-2 f(2)+--------------- \\ = & ---------+[a] f(a)-[a] f([a])\end{aligned}$

Terms start cancelling out,

We get,

$\begin{aligned} & -f(1)-f(2)-f(3)------------------- \\ - & ---f[a]+[a] f(a)\end{aligned}$

=$=[a] f(a)-(f(1)+f(2)+\cdots-\cdots-\cdots-\cdots-\cdots([a])$

Example 3: $\int_0^{\sqrt{2}}\left[x^2\right] d x$ is

1) $2-\sqrt{2}$
2) $2+\sqrt{2}$
3) $\sqrt{2}-$
4) $\sqrt{2}-2$

Solution

Fundamental Properties of Definite integration -

If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.

$\int_b^a f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x$

- wherein

$\begin{aligned} & \int_0^2\left[x^2\right] d x=\int_0^1\left[x^2\right] d x+\int_1^{\sqrt{2}}\left[x^2\right] d x \\ & \Rightarrow 0+\int_1^{\sqrt{2}} 1 d x=\sqrt{2}-1\end{aligned}$

Example 4: Choose the correct option?

1) $\int_{-a}^a f(x) d x=\int_{-a}^0 f(x) d x+\int_0^a f(x) d x$
2) $\int_a^c f(x) d x=\int_a^d f(x) d x+\int_d^c f(x) d x$; where $a<d<c$
3) $\int_a^c f(x) d x=\int_a^b f(x) d x+\int_b^c f(x) d x$; where $a<c<b$

4) All are true.

Solution

As we have learnt,

Fundamental Properties of Definite integration -

If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.

$\int_b^a f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x$

- wherein

It doesn't matter if b lies between a and c or not.


Example 5: The value of the integral $\int_{-2}^2 \frac{\sin ^2 x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}} d x$(where $[x]$ denotes the greatest integer less than or equal to (x) ) is :

1) $\sin 4$

2) 0

3) 4

4) $4-\sin 4$

Solution

Fundamental Properties of Definite integration -

If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.

$\int_b^a f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x$

- wherein


$\begin{aligned} & I=\int_2^{-2} \frac{\sin ^2 x}{\left|\frac{1}{7}\right|+\frac{1}{2}} d x \\ & =\int_2^0 \frac{\sin ^2 x}{\frac{1}{2}} d x+\int_0^{-2} \frac{\sin ^2 x}{-1+\frac{1}{2}} d x \\ & 2 \oint_2^0 \sin ^2 x d x 1-2 \int_0^{-2} \sin ^2 x d x \\ & P u t x=-p \Rightarrow d x=-d y \sin ^2(-p)=\sin ^2 p \\ & =2 \oint_0^2 \sin ^2 x d x+2 \int_2^0 \sin ^2 P d p\end{aligned}$

Summary

Definite integration is a powerful tool in calculus that allows us to calculate the area under a curve between two specific points. Piecewise integration is useful when the function is discontinuous, It provides a deeper understanding of mathematical ideas paramount for later developments in many scientific and engineering disciplines.

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Arrange the following Cobalt complexes in the order of incresing Crystal Field Stabilization Energy (CFSE) value. Complexes :  

\mathrm{\underset{\textbf{A}}{\left [ CoF_{6} \right ]^{3-}},\underset{\textbf{B}}{\left [ Co\left ( H_{2}O \right )_{6} \right ]^{2+}},\underset{\textbf{C}}{\left [ Co\left ( NH_{3} \right )_{6} \right ]^{3+}}\: and\: \ \underset{\textbf{D}}{\left [ Co\left ( en \right )_{3} \right ]^{3+}}}

Choose the correct option :
Option: 1 \mathrm{B< C< D< A}
Option: 2 \mathrm{B< A< C< D}
Option: 3 \mathrm{A< B< C< D}
Option: 4 \mathrm{C< D< B< A}

The type of hybridisation and magnetic property of the complex \left[\mathrm{MnCl}_{6}\right]^{3-}, respectively, are :
Option: 1 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 2 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 3 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 4 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 5 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 6 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 7 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 8 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 9 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 10 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 11 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 12 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 13 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 14 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 15 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 16 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
The number of geometrical isomers found in the metal complexes \mathrm{\left[ PtCl _{2}\left( NH _{3}\right)_{2}\right],\left[ Ni ( CO )_{4}\right], \left[ Ru \left( H _{2} O \right)_{3} Cl _{3}\right] \text { and }\left[ CoCl _{2}\left( NH _{3}\right)_{4}\right]^{+}} respectively, are :
Option: 1 1,1,1,1
Option: 2 1,1,1,1
Option: 3 1,1,1,1
Option: 4 1,1,1,1
Option: 5 2,1,2,2
Option: 6 2,1,2,2
Option: 7 2,1,2,2
Option: 8 2,1,2,2
Option: 9 2,0,2,2
Option: 10 2,0,2,2
Option: 11 2,0,2,2
Option: 12 2,0,2,2
Option: 13 2,1,2,1
Option: 14 2,1,2,1
Option: 15 2,1,2,1
Option: 16 2,1,2,1
Spin only magnetic moment of an octahedral complex of \mathrm{Fe}^{2+} in the presence of a strong field ligand in BM is :
Option: 1 4.89
Option: 2 4.89
Option: 3 4.89
Option: 4 4.89
Option: 5 2.82
Option: 6 2.82
Option: 7 2.82
Option: 8 2.82
Option: 9 0
Option: 10 0
Option: 11 0
Option: 12 0
Option: 13 3.46
Option: 14 3.46
Option: 15 3.46
Option: 16 3.46

3 moles of metal complex with formula \mathrm{Co}(\mathrm{en})_{2} \mathrm{Cl}_{3} gives 3 moles of silver chloride on treatment with excess of silver nitrate. The secondary valency of CO in the complex is_______.
(Round off to the nearest integer)
 

The overall stability constant of the complex ion \mathrm{\left [ Cu\left ( NH_{3} \right )_{4} \right ]^{2+}} is 2.1\times 10^{1 3}. The overall dissociation constant is y\times 10^{-14}. Then y is ___________(Nearest integer)
 

Identify the correct order of solubility in aqueous medium:

Option: 1

Na2S > ZnS > CuS


Option: 2

CuS > ZnS > Na2S


Option: 3

ZnS > Na2S > CuS


Option: 4

Na2S > CuS > ZnS


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