Select Page

Definite Integrals

In Geometry, we know how to calculate the area of a triangular plot of land.

We can also find the area of the plot, if it is in the shape of a rectangle or a Square.

All the shapes mentioned above (triangle, rectangle, square) are regular shapes, whose area can be found out by the known results called formulas.

Now, if we are given a shape which is not regular or standard one, then how do we find its area? We try to approximate its area by associating it with regular shapes.

For finding the area of such irregular shapes, area enclosed between two geometrical shapes we make use of Definite Integrals.

Here, we introduce the concept of Definite Integrals.

  • A definite integral is denoted by\int_{a}^{b}  f(x)dx where a is called the lower limit of the integral & b is called the upper limit of the integral.

 

  • \int_{a}^{b}  f(x)dx denotes the area of the region bounded by the curve y = f(x), the ordinates x = ax = band the x-axis.

  • Since the definite integral denotes the area enclosed, hence its value is always definite or fixed. So, it is called definite integral.

 

  • Relation between Indefinite & definite integral.

If f(x) be a continuous function defined on the closed interval [ab] & F(x) is its antiderivative i.e.\int_{}^{}  f(x)dx = F(x) , then \int_{a}^{b}  f(x)dx = [F(x)]_{a}^{b}  = F(b) – F(a)

The above statement is called the Second fundamental Theorem of integral calculus.

 

Note: While evaluating a definite integral the Constant of Integration, C disappears at the end and hence its effect is nullified. So, we do not write the constant of integration while evaluating a definite integral.

Examples

Now let’s consider some examples on definite integrals .

Example 1: Evaluate the definite integral:\int_{0}^{2}(x^2 +1)dx

Let F(x) = \int(x^2 + 1)dx = \frac{x^{2+1}}{2+1} + x = \frac{x^{3}}{3} +x

\int_{0}^{2}(x^2 +1)dx = [F(x)]_{0}^{2} = F(2) – F(0)

\left ( \frac{2^3}{3} +2 \right ) – \left ( \frac{0^3}{3} +0 \right ) = \left ( \frac{8}{3} +2 \right ) – 0 = \frac{14}{3}

Note: The constant of Integration is not written here, as it does not change the final value of the definite integral.

 

Example 2Evaluate: \int_{-1}^{1} (e^x +x)dx

Let F(x) = \int (e^x +x) dx = e^x  + \frac{x^{1+1}}{1+1}  = e^x  + \frac{x^{2}}{2}

\int_{-1}^{1}(e^x +x)dx = [F(x)]_{-1}^{1} = F(1) – F(-1)

\left ( e^1+\frac{(1)^{2}}{2} \right ) – \left ( e^{-1}+\frac{(-1)^{2}}{2} \right ) = e +\frac{1}{2} – e^{-1} – \frac{1}{2}

e – e^{-1} = e – \frac{1}{e}

 

Example 3: Evaluate:\int_{1}^{5} x^2\left ( 1 - \frac{1}{x^3} \right )dx

Let F(x) = \int x^2\left ( 1 - \frac{1}{x^3} \right )dx = \int\left ( x^2 - \frac{1}{x} \right )dx

=\int x^2dx – \int\frac{1}{x} dx = \frac{x^{2+1}}{2+1}  – log |x| = \frac{x^3}{3} – log|x|

\int_{1}^{5} x^2 \left ( 1 - \frac{1}{x^3} \right )dx = [F(x)]_{1}^{5} = F(5) – F(1)

\left ( \frac{5^3}{3}-log|5| \right ) – \left ( \frac{1^3}{3}-log|1| \right ) = \left ( \frac{125}{3}-log|5| \right ) – \left ( \frac{1}{3}-log|1| \right )

\left ( \frac{125}{3}-\frac{1}{3} \right ) – log|5| + log|1| = \frac{124}{3} – log|5| + 0 = \frac{124}{3} – log|5|

 

Example 4:  Evaluate: \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{3+4Cosx}{Sin^2x}dx

Let F(x) = \int\frac{3+4Cosx}{Sin^2x}dx = \int \left ( \frac{3}{Sin^2x} + 4 \frac{Cosx}{Sin^2x} \right )dx 

=\int (3Cosec^2x + 4CosecxCotx)dx = 3\int Cosec^2x dx + 4 \int CosecxCotx

-3Cotx – 4Cosecx

\int_{0}^{\frac{\pi}{2}} \frac{3+4Cosx}{Sin^2x}dx = [F(x)]_{0}^{\frac{\pi}{2}} = F\left ( \frac{\pi}{2} \right ) – F\left ( \frac{\pi}{4} \right )

 

\left ( -3Cot\frac{\pi}{2} - 4Cosec\frac{\pi}{2} \right ) – \left ( -3Cot\frac{\pi}{4} - 4Cosec\frac{\pi}{4} \right )

= (-3(0)-4(1)) – \left ( -3(1)-4(\sqr2) \right )

= (0 – 4) – \left ( -3-4\sqr{2} \right ) = 4 + 3 +4\sqr{2} = -1 +4\sqr{2}

 

Example 5: Evaluate \int_{1}^{4}\left ( 3x^2 -2x +7\sqr{x} \right )dx

Let F(x) =\int\left ( 3x^2 -2x +7\sqr{x} \right )dx

= 3\int x^2 dx – 2\int xdx + 7 \intx^{1/2}dx = 3 \frac{x^{2+1}}{2+1} – 2\left ( \frac{x^{1+1}}{2} \right ) + 7 \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} +C

x^3 –x^2 +\frac{14}{3} x^{\frac{3}{2}} = x^3 –x^2 + \frac{14}{3} x\sqr{x}

\int_{1}^{4}\left ( 3x^2-2x+7\sqr{x} \right )dx = [F(x)]_{1}^{4} = F(4) – F(1)

\left ( 4^3 - 4^2 +\frac{14}{3}4\sqr{4} \right ) – \left ( 1^3 - 1^2 +\frac{14}{3}(1)\sqr{1} \right )

\left ( 64 - 16 +\frac{112}{3}\right ) – \left ( 0 +\frac{14}{3}\right ) = 48 + 98 = \frac{144-98}{3} = \frac{46}{3}

 

Check Point

Evaluate the following definite integrals:

  1. \int_{0}^{1} (x + 1)dx
  2. \int_{0}^{\frac{\pi}{4}}  \left ( 7Cosx - 3Sec^2x \right )dx
  3. \int_{0}^{1} \left ( 7e^x + 34x^{16}+17 \right )dx
  4. \int_{1}^{2}(x – 1)(x – 2)dx
  5. \int_{1}^{4}\left ( \sqr{x} - \frac{1}{\sqr{x}} \right )dx
Answer Key
  1. \int_{0}^{1} (x + 1)dx = \frac{3}{2}
  2. \int_{0}^{\frac{\pi}{4}}  \left ( 7Cosx - 3Sec^2x \right )dx = 
  3. \int_{0}^{1} \left ( 7e^x + 34x^{16}+17 \right )dx = 7e +19
  4. \int_{1}^{2}(x – 1)(x – 2)dx = -\frac{1}{6}
  5. \int_{1}^{4}\left ( \sqr{x} - \frac{1}{\sqr{x}} \right )dx = \frac{17}{2}

Give Your Child The eTutorWorld Advantage

Research has proven that personal online tutoring not just cements school learning, it helps build student confidence. eTutorWorld provides the best K-12 Online Tutoring Services so you can learn from the comfort and safety of your home at an affordable cost.

Be it an exam, class test or a quiz, eTutorWorld’s Math, Science and English tutors are responsible for your academic progress. Meet your personal coach at your convenient day and time to get help for Grade 3-12 Math, Science and English subjects and AP, SAT, SSAT and SCAT Test Prep help and test practice. All our tutors are graduates and bring with them years of teaching experience to the tutoring lessons.

Our ‘Learning by Design’ methodology makes sure that each student is at the center of the teaching-learning process. All tutoring sessions start with a question to the student ‘What do you want to learn today?’ Hence, tutors diagnose your skills and recognize your requirements before the actual tutoring happens. Post every tutoring session, an individualized worksheet is emailed to the student to assimilate learned concepts. Regular formative assessments are used to evaluate a student’s understanding of the subject.

The state of the art technology used is stable, user friendly and safe. All you need is a computer or a tablet and an internet connection. The easy-to-use web conferencing software requires a one time download, using which the student can talk and chat to the tutor, annotate on an interactive shared whiteboard or even share documents, assignments or worksheets.

All tutoring sessions are recorded and made available for a month so you can review concepts taught.

Email or call our support team with any issues or questions – we are here for you 24X7.

Also, download free printable math and science worksheets in pdf format and solve SCAT and SSAT Practice Tests online. Sign up for a Free Trial Lesson Today!

Thousands have taken the eTutorWorld Advantage – what are you waiting for?

© 2019 eTutorWorld - Online Tutoring and Test Prep | All rights reserved