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Definite Integration

Learn definition of the definite integral with all the steps. Evaluate definite integration as a limit of riemann Sum & sigma notation calculus formula.

The Definition of Definite Integral

Let  \(f\) be a function defined in [a, b], we divide the interval [a, b] into \(n\) subintervals; whose width is 

\(\Delta x=\dfrac{b-a}{n}\)

Let \(x_0(a), \;x_1,\,x_2,......x_{n-1},\;x_n(b)\) be the end points of the subintervals \([x_0,\;x_1],\;[x_1,\;x_2],.......[x_{n-1,\;x_n}]\). Let \(x_1^*,\;x_2^*.......x_n^*\) be sample points in these subintervals such that \(x_1^*\in\;\;[x_{i-1},\;x_i]\).

  • The definite integral of \('f'\) from \('a'\)  to \('b'\) is defined as

\(\displaystyle\int \limits^b_a f(x)dx=\lim\limits _{n\to\infty}\sum^n_{i=1}(f(x_i^*)\Delta x)\)

  • This assumes that the limit exists. If this limit can be found we say \('f'\) integrable in [a, b].
  • \('a'\) is called lower limit, \('b'\) the upper limit of integral.
  • \('f'\) is called integral function and process of calculating integral is called integration.

Illustration Questions

Express the limit as a definite integral on given interval. \(\lim\limits _{n\to\infty\,\,\,}\sum\limits^n_{i=1}\left(\dfrac{2+x_i}{5+3x_i}\right)\Delta x\), \(x\in[5,\,9]\)

A \(\displaystyle\int \limits^{13}_{11} \dfrac{2-x}{5+x}dx\)

B \(\displaystyle\int \limits^{9}_{5} \left(\dfrac{2+x}{5+3x}\right)dx\)

C \(\displaystyle\int \limits^{12}_{7} \dfrac{2x+1}{x-2}dx\)

D \(\displaystyle\int \limits^{50}_{7} x\;dx\)

×

\(\displaystyle\int \limits^b_a f(x)dx=\lim\limits _{n\to\infty}\,\,\sum^n_{i=1}\left(f(x_i^*)\Delta x\right)\,\,\,\,\to x_i^*\text{ can be} \,x_i\,or\;x_{i-1}\).

Here,  \(a=5,\;b=9,\;f(x_i)=\dfrac{2+x_i}{5+3x_i}\)

\(\therefore\,\lim\limits _{n\to\infty}\,\,\sum\limits^n_{i=1}\left(\dfrac{2+x_i}{5+3x_i}\right)\Delta x\)        ( Replace \(\left(\lim\sum\right)\) by \(\int\))

\(=\displaystyle\int \limits^{9}_{5} \left(\dfrac{2+x}{5+3x}\right)dx\)          (\(\Delta x\) is replaced by \(dx\))

Express the limit as a definite integral on given interval. \(\lim\limits _{n\to\infty\,\,\,}\sum\limits^n_{i=1}\left(\dfrac{2+x_i}{5+3x_i}\right)\Delta x\), \(x\in[5,\,9]\)

A

\(\displaystyle\int \limits^{13}_{11} \dfrac{2-x}{5+x}dx\)

.

B

\(\displaystyle\int \limits^{9}_{5} \left(\dfrac{2+x}{5+3x}\right)dx\)

C

\(\displaystyle\int \limits^{12}_{7} \dfrac{2x+1}{x-2}dx\)

D

\(\displaystyle\int \limits^{50}_{7} x\;dx\)

Option B is Correct

Independency of Variable in Definite Integration

  • \(\displaystyle\int\limits^b_af(x)dx=\displaystyle\int\limits^b_af(t)dt=\int\limits^b_af(z)dz\)

The definite integral does not depend upon variable of integration \(x\), we can replace \(x\) by \(t\) or \(z\), without changing the function \('f'\).

Illustration Questions

If  \(\displaystyle\int\limits^\pi_0(sin\,x)dx=2\)  then the value of  \(\displaystyle\int\limits^\pi_0(sin\,t)dt\)  is

A 2

B –2

C 18

D –44

×

\(\displaystyle\int\limits^b_af(x)dx=\displaystyle\int\limits^b_af(t)dt\)

\(\displaystyle\int\limits^\pi_0(sin\,x)dx=\int\limits^\pi_0(sin\,t)dt\)         (Only \(x\) is replaced by \('t'\))

= 2

 

If  \(\displaystyle\int\limits^\pi_0(sin\,x)dx=2\)  then the value of  \(\displaystyle\int\limits^\pi_0(sin\,t)dt\)  is

A

2

.

B

–2

C

18

D

–44

Option A is Correct

Riemann Sum

The sum  \(\sum\limits^n_{i=1}\,f(x_i^*)\,\Delta x\) where \(\Delta x=\dfrac{b–a}{n}\) is called the Riemann Sum,

after a German  Mathematician Bernhard Riemann.

  • If \('f'\) is a positive function then Riemann sum is sum of areas of rectangles which approximate the area bounded by a positive function.
  • If \(f(x)\geq0\)

\(\displaystyle\int\limits^b_a f(x)\,dx\) is the limit of this sum which is the exact area under the curve \(y=f(x)\) from \(a\) to \(b\).

  • If \('f'\) takes both positive and negative values then Riemann sum is sum of areas of rectangle that lie above \(x\) axis and negative of area of rectangles that lie below \(x\) axis.

Riemann sum = shaded area  \((A_1)\)– unshaded area \((A_2)\)

 

In terms of definite integral,

\(\displaystyle\int\limits^b_a f(x)\,dx=A_1–A_2\)

Illustration Questions

Let \(f(x)=x^2–3x\), \(0\leq x\leq 4\), find the Riemann sum with \(n=4\) taking sample points to be right end points.

A 17

B 0

C 9

D –8

×

Riemann Sum (R.S) = \(\sum\limits^4_{i=1} f(x_i^*)\,\Delta x\)

\(\Delta x=\dfrac{4-0}{4}=\dfrac{4}{4}=1\)      \(\;x_i^*=x_i\to\)  right end points

 

\(\therefore\,R.S=f(x_1)\Delta x+f(x_2)\Delta x+f(x_3)\Delta x+f(x_4)\Delta x\)

\(=f(1)×1+f(2)×1+f(3)×1+f(4)×1\)

\(=(1–3)+(4–6)+(9–9)+(16–12)\)

\(=–2–2+0+4=0\)

This means that area above \(x\) - axis is same as area below \(x\) - axis.

image

Let \(f(x)=x^2–3x\), \(0\leq x\leq 4\), find the Riemann sum with \(n=4\) taking sample points to be right end points.

A

17

.

B

0

C

9

D

–8

Option B is Correct

Evaluation of some Standard Sequences

The sigma notation is often used to conveniently write the sequences and series.

Consider 

\(\sum\limits^n_{i=1}a_i=a_1+a_2+a_3+.......a_n\)

i.e.  write the term with values of variable of summative and then insert + sign.

(1)  \(\sum\limits^n_{i=1}c=cn\) (where c is a constant independent of \(n\))

(2)  \(\sum\limits^n_{i=1}ca_i=c\sum\limits^n_{i=1}a_i\)

(3)  \(\sum\limits^n_{i=1}a_i+b_i=\sum\limits^n_{i=1}a_i+\sum\limits^n_{i=1}b_i\)

(4)  \(\sum\limits^n_{i=1}a_i-b_i=\sum\limits^n_{i=1}a_i-\sum\limits^n_{i=1}b_i\)

 

 

(1)  \(\sum\limits^n_{i=1}\,i=1+2+3+4+.....n=\dfrac{n(n+1)}{2}\) 

(Sum of first \(n\) natural numbers is \(\dfrac{n(n+1)}{2}\))

(2)  \(\sum\limits^n_{i=1}\,i^2=1^2+2^2+3^2+.....n^2=\dfrac{n(n+1)(2n+1)}{6}\)

(Sum of squares of first \(n\) natural numbers is \(\dfrac{n(n+1)(2n+1)}{6}\))

(3)  \(\sum\limits^n_{i=1}\,i^3=1^3+2^3+3^3+.....n^3=\left(\dfrac{n(n+1)}{2}\right)^2\)

Sum of cubes of first \(n\) natural numbers is \(\left(\dfrac{n(n+1)}{2}\right)^2\).

Illustration Questions

Find the value of the sum \(S=1^2+2^2+.....15^2\) .

A 1240

B –89

C 700

D 26

×

\(1^2+2^2+......15^2=\sum\limits^{15}_{i=1}\,i^2\)

\(=\sum\limits^{15}_{i=1}\,i^2=\dfrac{15×16×31}{6}\)  \(\bigg[\) Use \(\dfrac{n(n+1)(2n+1)}{6}\) with \(n=15\) \(\bigg]\)

\(=1240\)

Find the value of the sum \(S=1^2+2^2+.....15^2\) .

A

1240

.

B

–89

C

700

D

26

Option A is Correct

The Riemann Sum and Definite Integrals

  • \(\displaystyle\int\limits^b_af(x)dx=\lim\limits_{n\to \infty}\,\left(\sum\limits^n_{i=1}\,f(x_i)\Delta x\right) \)

where \(x_i\) is the right end point (Normally we use \(x_i^*=x_i\) for convenience)

  • \(\Delta x=\dfrac{b–a}{n},\;x_i=a+i\Delta x\)
  • Express the entire expression \(E \) in terms of \(x\) and then take the limit.

Illustration Questions

Evaluate \(\displaystyle\int\limits^4_2(x^2+x)dx\) by calculating the limit of the sum.

A \(\dfrac{74}{3}\)

B 1

C –18

D \(\dfrac{8}{9}\)

×

\(\displaystyle\int\limits^4_2(x^2+x)dx=\lim\limits_{n\to \infty}\,\sum\limits^n_{i=1}\,f(x_i)\Delta x\)

\(\Delta x=\dfrac{4–2}{n}=\dfrac{2}{n},\;\;x_i=2+\dfrac{2i}{n}\)

\(\displaystyle\int\limits^4_2(x^2+x)dx=\lim\limits_{n\to \infty}\,\sum\limits^n_{i=1}\,f\left(2+\dfrac{2i}{n}\right)×\dfrac{2}{n}\)

\(=\lim\limits_{n\to \infty}\,\sum\limits^n_{i=1}\,\left[\left(2+\dfrac{2i}{n}\right)^2+\left(2+\dfrac{2i}{n}\right)\right]×\dfrac{2}{n}\)

\(=\lim\limits_{n\to \infty}\,\sum\limits^n_{i=1}\,\left(4+\dfrac{4i^2}{(n)^2}+\dfrac{8i}{n}+2+\dfrac{2i}{n}\right)×\dfrac{2}{n}\)

\(=\lim\limits_{n\to \infty}\,\left(\sum\limits^n_{i=1}\,4+\dfrac{4}{n^2}\sum\limits^n_{i=1} i^2+\dfrac{8}{n}\sum\limits^n_{i=1} i+\sum\limits^n_{i=1}2+\dfrac{2\sum\limits^n_{i=1} i}{n}\right)×\dfrac{2}{n}\)

\(=\lim\limits_{n\to \infty}\left(4n+\dfrac{4n(n+1)(2n+1)}{6n^2}+\dfrac{8n(n+1)}{2n}+2n+\dfrac{2n(n+1)}{2n}\right)×\dfrac{2}{n}\)

\(=\lim\limits_{n\to \infty}\left(8+\dfrac{4}{3}\dfrac{(n+1)(2n+1)}{n^2}+\dfrac{8(n+1)}{n}+4+\dfrac{2(n+1)}{n}\right)\)

\(=\lim\limits_{n\to \infty}\left[8+\dfrac{4}{3}\left(\left(1+\dfrac{1}{n}\right)\left(2+\dfrac{1}{n}\right)\right)+8\left(1+\dfrac{1}{n}\right)+4+2\left(1+\dfrac{1}{n}\right)\right]\)

\(=8+\dfrac{8}{3}+8+4+2\)        \(\left(n\to \infty \Rightarrow\,\dfrac{1}{n}\to 0\right)\)

\(=\dfrac{74}{3}\)

Evaluate \(\displaystyle\int\limits^4_2(x^2+x)dx\) by calculating the limit of the sum.

A

\(\dfrac{74}{3}\)

.

B

1

C

–18

D

\(\dfrac{8}{9}\)

Option A is Correct

Illustration Questions

Evaluate  \(\sum\limits^{20}_{i=1}\;(2i^2+3i)\) .

A 5000

B 6370

C 12

D –89

×

\(\sum\limits^{20}_{i=1}\;(2i^2+3i)\) \(=\sum\limits^{20}_{i=1}\;2i^2+\sum\limits^{20}_{i=1}\;3i\)

 

\(\bigg(\sum (a_i+b_i)=\sum a_i+\sum b_i\bigg)\)

\(=2\sum\limits^{20}_{i=1}\;i^2+3\sum\limits^{20}_{i=1}\;i\)        \(\bigg(\sum c_ia=a\sum c_i\bigg)\)

 

\(=\dfrac{20n(n+1)(2n+1)}{6}+\dfrac{3n(n+1)}{2}\) with \(n=20\)

\(=2×\dfrac{20×21×41}{6}+\dfrac{3×20×21}{2}\)

\(=5740+630\)

\(=6370\)

Evaluate  \(\sum\limits^{20}_{i=1}\;(2i^2+3i)\) .

A

5000

.

B

6370

C

12

D

–89

Option B is Correct

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