Area bounded by non negative function

Area Bounded By Non Negative Function

The Definite Integration of a non negative function f(x) with respect to ‘x’ from ‘a’ to ‘b’ gives the area of the bounded region abBAa (given in above figure). Now let f(x) is a countinuous non negative function on an interval [a, b], such that f(x)≥0. Then the area of the closed region bounded by the function f(x) from ‘a’ to ‘b’ with respect to ‘x’ is equal to the definite integral \int ^{b}_{a}f\left( x\right) dx.

Now for the best understanding of the topic you are highly recommended to attempt at least five questions based on the theory that you have learned.

Example: find the area of the region bounded by the curve x²-2x+4, x-axis and line segments x = -1 and x = 3.
Solution:
\begin{aligned}\int ^{3}_{-1}\left( x^{2}-2x+4\right) dx=\int ^{3}_{-1}x^{2}dx-2\int ^{3}_{-1}xdx+4\int ^{3}_{-1}dx\\ \Rightarrow \left( \dfrac{x^{3}}{3}\right) _{-1}^{3}-2\left( \dfrac{x^{2}}{2}\right) _{-1}^{3}+4\left( x\right) _{-1}^{3}=\dfrac{52}{3}\end{aligned}

Now You People are Advised to Attempt the following unsolved problems for better understanding this topic.

Question: Find the area of the region bounded by the function Cos-1(cos x), x-axis on the interval [0, 2π].

Question: Let f(x)=cos-1(cos x)-sin-1(sin x) is a continuous function. Find the area of the region bounded by the function f(x) and x-axis in [0, 2π].

Question: Find the area of the region bounded by the function f(x)=x|sin x|and x-axis in the interval [0, 2π].

Question: Find the area of the region bounded by the curve f(x)=tan-1(x)+ln(x), x-axis, and the straight line segments x=1 and x=2.

Question: Let f(x) be a function defined as f(x)=max{|x-2|, (4-x2), (x-2)1/3}, x belongs to [-2, 4]. Now find the area bounded by the curve and x-axis.

So this is it from this tutorial. Hoping you people have attempted all the unsolved problems that have written above. In the next tutorial, we will discuss the topic “area of the region bounded by the non-positive continuous function”.

Leave a Comment

Your email address will not be published. Required fields are marked *