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Integration- Area Under Curve

Updated: Jun 22, 2021



In this note, you will learn:


· Area of region above the x-axis

· Area of region below the x-axis



From the previous Integration article, we have learnt about Definite Integrals and we know that definite integrals are the areas bounded the specific upper and lower limits of the independent variable.



Today, we will be learning about the different ways a region can be bounded on a graph, and how can we solve for the value of the area. Let’s begin!

Area of region above the x-axis


Since we know that definite integrals represent the area under the curve, an area of a region bounded above the x-axis will look something like this:



As you see from the curve in the diagram above, the area is bounded above the x-axis, in between the x-axis and the curve and between the limits of a and b.



In order to solve for these types of questions, we will have to apply the concept of definite integrals we learnt back in the previous article.



Given that y = 4x^4, and a = 2 and b = 5, we can find the area of the shaded region with the following steps:


5

Area of shaded region = ∫2 4x^4 dx

5

= [(4/5) x^5]2


= [(4/5) (5)^5] – [(4/5) (2)^5]

= 2500 – 25.6

= 2474.4

= 2470 units^2 (3 s.f)



Note: Since no units were specified in the question, hence we used the general term “units” to quantify the value of the answer. However, we are aware that the standard units of measurement of area are units squared, therefore we write it as such: units^2.



Area of region below the x-axis


Next, let’s take a look at the following curve in the diagram shown below:




The diagram above shows a function y = f(x), where f(x) < 0 for a < x < b. The evaluation of the definite integral from x = a to x = b, i.e. ∫ba f(x) dx will result in a negative value, which indicates that the region lies below the x-axis. Since values of area are always positive, then the results will be as follows:



Area of region bounded by the curve y = f(x), the x-axis and the lines x = a and x = b

is given by: |∫ba f(x) dx|, where f(x) ≤ 0 for a ≤ x ≤ b


Let’s take a look at an example of the application below:

The diagram shows part of the curve y = (x – 3) (x – 2). Find the area of the shaded region.




Solution:

3

Area of shaded region = |∫1 (x – 3) (x – 2) dx|

3

= |∫1 (x2 – 5x + 6) dx|

3

= | [(1/3) x^3 – (5/2) x^2 +6x]1 |

= | [(1/3) (3)^3 – (5/2) (3)^2 +6(3)] – [(1/3) (1)^3 – (5/2) (1)^2 +6(1)] |

= | [(27/3) – (45/2) + 18] – [(1/3) – (5/2) + 6] |

= | (9/2) – (23/6) |

= 2/3 units^2


And that’s all for today, students! Math Lobby hopes that after this article, you have a clear understanding on how to find the area under curve, and is equipped with the necessary skills to deal with questions that involve finding the area under curve bounded above/below the x-axis.



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