Integration is sometimes called antidifferentiation, as it is the opposite process of differentiation. When we find an indefinite integral, we find a function with an arbitrary constant, C. Whereas, when we find a definite integral, we find a numerical value. Definite integration has many applications. For the examination, we use it to find displacement in questions about kinematics, probabilities with continuous random variables and it is linked to work on finding areas and volumes with integration.
Key Concepts
On this page, you should learn about
definite integrals
\(\int_{a}^{b} f'(x)\: dx=\: f(b)-f(a)\)
Essentials
The following videos will help you understand all the concepts from this page
What and Why
In the following video, we are going to look at what definite integration is (comparing to indefinite integration) and what we use it for. We'll look at some of the many applications of definite integration.
How to do it
The process of finding definite integrals is relatively straightforward, but it is important to carefully use the correct notation. In the following video, we will look at two examples: \(\int_{\frac{\pi}{2}}^{\pi} cosx\: dx\) and \(\int_{1}^{e} \frac{1}{x}\: dx\)
b) f(x - 2) translates the graph of f(x) 2 units to the right. This is the same as \(\int _{ 0 }^{ 2 }{ f(x)dx } \)
c) f(x) + 2 translates the graph 2 units up. It creates a region the same as \(\int _{ -2 }^{ 2 }{ f(x)dx=5 } \) , but with a rectangle below. The area of the rectangle is 8.
Exam-style Questions
Question 1
Given that \(\int _{ 4 }^{ 8 }{ \frac { 1 }{ 2x-4 } dx= } ln\sqrt { a } \) , find the value of a