chevron icon chevron icon chevron icon chevron icon

Applications Of Integration

In this article, we will be learning about the different applications of integration. We will cover the following sub-topics:

  • Evaluating definite integrals
  • Properties of definite integrals

Evaluating Definite Integrals

Introduction

The notation \(\int_a^bf(x) \,dx \) denotes \(F(b) – F(a)\) where \(F\) is the antiderivative of \(f\).

Here \(a\) is called the lower bound and \(b\) is the upper bound.

We simply write it as, 

\(\small{\begin{align*} \int_a^b f (x) \; dx &= \bigg [\,F (x)\,\bigg]_a^b \\ \\ &= F(b)- F(a)  \end{align*}}\)

\(F(a)\) is obtained by substituting \(x = a \) into the function \(F\), while \(F(b)\) is obtained by substituting \( x = b \) into the function \(F\).

 

How is the notation \(\small{\displaystyle{\int_a^bf(x) \,dx}}\) read?

Integrate \(f(x)\) from \(a\) to \(b\) (from lower bound to upper bound), with respect to \(x\).

It represents the area under the curve \(f(x)\) from \(a\) to \(b\).

Properties Of Definite Integrals

First Property Of Definite Integrals

The definite integral of the scalar multiple of a function can be evaluated as such:

\(\small{\begin{align*}  \int_a^b kf (x) \;dx =  k \bigg[\;F (x)\;\bigg]_a^b \end{align*}}\)

 

Second Property Of Definite Integrals

The definite integral of the sum or difference of functions can be evaluated as such:

\(\small{\begin{align*}  \int_a^b f (x) \;dx \pm \int_a^b g(x) \;dx =  \int_a^b f (x) \pm g(x) \;dx \end{align*}}\)

Keep in mind that the lower bound \(a\) and upper bound \(b\) must be the same for both integrals. 

 

Question 1:

Evaluate the following integrals.

\(\small{\mathrm{A.}\; \displaystyle{\int_1^2\,1+x+x^2\;dx}}\)

Solution:

\(\small{\begin{align*} \int_1^21+x+x^2\;dx &= \bigg[\; x+\frac{x^2}{2}+\frac{x^3}{3} \;\bigg]_1^2 & &\text{First, integrate the expression.} \\ \\ &= \bigg(\;2+\frac{4}{2}+\frac{8}{3}\;\bigg) - \bigg(\;1+ \frac{1}{2}+\frac{1}{3}\;\bigg) & &\text{Substitute} \; x = 2\; \text{and}\; x = 1. \\ \\ &= 6\frac{2}{3} - 1\frac{5}{6} & &\text{Subtract the expressions.} \\ \\ &= 4\frac{5}{6} & &\text{Evaluate.} \end{align*}}\)

 

\(\small{\mathrm{B.}\; \displaystyle{\int_0^{\tfrac{\pi}{6}} cos\;3x \;dx}}\)

Solution:

\(\small{\begin{align*} \int_0^{\tfrac{\pi}{6}} cos\;3x \;dx &= \bigg[\; \frac{sin\,3x}{3} \;\bigg]_0^{\tfrac{\pi}{6}} \\ \\ &= \bigg(\; \frac{sin\,\frac {\pi}{2}} {3} \;\bigg) - \bigg(\; \frac {sin\,0} {3} \;\bigg)\\ \\ &= \frac {1}{3} - 0 \\ \\ &= \frac {1}{3} \end{align*}}\)

 

\(\small{\mathrm{C.}\; \displaystyle{\int_2^3 \frac{2}{2x-3} \;dx}}\)

Solution:

\(\small{\begin{align*} \int_2^3 \frac{2}{2x-3} \;dx &= \bigg[\; \frac{2\,\ln (2x-3)}{2} \;\bigg]_2^6 \\ \\ &= \bigg[\; \ln (2x-3) \;\bigg]_2^6 \\ \\ &= \big(\; \ln (2\times6-3) \;\big) - \big(\; \ln (2\times2-3) \;\big) \\ \\ &= \big(\; \ln (12-3) \;\big) - \big(\; \ln (4-3) \;\big) \\ \\ &= \ln 9 - \ln 1 \\ \\ &= \ln 9 - 0 \\ \\ &= \ln 9 \end{align*}}\)

 

Question 2:

Evaluate \(\small{\displaystyle{\int_1^2 1 +x^2 \;dx}}\)

 

Solution:

\(\small{\begin{align*} \int_1^2 1 +x^2 \;dx &= \bigg[\; x+\frac{x^3}{3}\;\bigg]_1^3 \\ \\ &= \bigg(\; 3+\frac{27}{3}\;\bigg)-\bigg(\; 1+\frac{1}{3}\;\bigg) \\ \\ &= 12 -1\frac{1}{3} \\ \\ &= 10\frac{2}{3} \end{align*}}\)

 

Question 3:

Evaluate \(\small{\displaystyle{\int_{\tfrac{\pi}{12}}^{\tfrac {\pi}{6}}3\;sec^2\,2x\;dx }}\), leaving your answer in exact form.

Solution:

\(\small{\begin{align*} \int_{\tfrac{\pi}{12}}^{\tfrac {\pi}{6}}3\;sec^2 2x\;dx &= \bigg[\; \frac{3 \;tan \;2x}{2} \;\bigg]_{\tfrac{\pi}{12}}^{\tfrac {\pi}{6}} \\ \\ &= \bigg(\; \frac{3 \;tan \;{\frac{\pi}{3}}}{2} \;\bigg) - \bigg(\; \frac{3 \;tan \;{\frac{\pi}{6}}}{2} \;\bigg) \\ \\ &= \frac { 3\sqrt{3} } {2} - \frac{3}{2} \cdot \frac{\sqrt{3}}{3} \\ \\ &= \frac{2\sqrt{3}}{2} \\ \\ &= \sqrt{3} \end{align*}}\)

Third Property Of Definite Integrals

If the lower bound and upper bound are equal in value, the result of the definite integral is \(0\).

\(\small{\begin{align*} \int_ a^a  f(x)\; dx &=  \bigg[\;F(x)\;\bigg]_a^a \\ \\ &= F(a) - F(a) \\ \\ &= 0 \end{align*}}\)

Fourth Property of Definite Integrals

If the lower bound and upper bound are swapped, the result of the new definite integral is equal in value but opposite in sign (i.e., positive vs. negative) compared to the original definite integral.

\(\small{\begin{align*} \int_ b^a  f(x) \;dx &=  \bigg[\;F(x)\;\bigg]_b^a \\ \\ &= F(a) - F(b) \\ \\ &= -\bigg(\;F(b) - F(a)\;\bigg) \\ \\ &= -\bigg[\;F(x)\;\bigg]_a^b \\ \\ &= -\int_ a^b  f(x)\, dx \end{align*}}\)

Fifth Property of Definite Integrals

For definite integrals of the same function, where the upper bound of the first integral is equal in value to the lower bound of the second integral:

\(\small{\begin{align*} \int_ a^b  f(x) \;dx + \int_ b^c  f(x) \;dx = \int_ a^c  f(x) \;dx \end{align*}}\)

Graphically,

Conclusion

In this article, we learned about definite integrals and how they are different from indefinite integrals. We also learned the five different properties associated with definite integrals.

Keep improving! Keep learning!
 

Continue Learning
Introduction To Differentiation Applications Of Differentiation
Differentiation Of Exponential And Logarithmic Functions Integration Techniques
Applications Of Integration  
Resources - Academic Topics
icon expand icon collapse Primary
icon expand icon collapse Secondary
icon expand icon collapse
Book a free product demo
Suitable for primary & secondary
select dropdown icon
Our Education Consultants will get in touch with you to offer your child a complimentary Strength Analysis.
Book a free product demo
Suitable for primary & secondary
Claim your free demo today!
Claim your free demo today!
Arrow Down Arrow Down
Arrow Down Arrow Down
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
Geniebook CTA Illustration Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Geniebook CTA Illustration Geniebook CTA Illustration
Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Get a free diagnostic report of your child’s strengths & weaknesses!
Arrow Down Arrow Down
Arrow Down Arrow Down
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
Gain access to 300,000 questions aligned to MOE syllabus
Trusted by over 220,000 students.
Trusted by over 220,000 students.
Arrow Down Arrow Down
Arrow Down Arrow Down
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
media logo
Geniebook CTA Illustration
Geniebook CTA Illustration
Geniebook CTA Illustration
icon close
Default Wrong Input
Get instant access to
our educational content
Start practising and learning.
No Error
arrow down arrow down
No Error
*By submitting your phone number, we have
your permission to contact you regarding
Geniebook. See our Privacy Policy.
Success
Let’s get learning!
Download our educational
resources now.
icon close
Error
Error
Oops! Something went wrong.
Let’s refresh the page!