f(x)≤0 Area Bounded By Non Positive Function

Area Bounded By Non Positive Function

The Definite Integration of a non positive function f(x) with respect to ‘x’ from ‘a’ to ‘b’ gives the area of the shaded region, given in above figure. Let f(x) be a non positive countinuous 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 Mod of the definite integral \int ^{b}_{a}f\left( x\right) dx.

Now for the best understanding of the topic. You are highly recommended to take a look at the solved examples given below and attempt at least five questions based on the theory that you have learned.

Example: Find the area of the region bounded by the function y =log1/4(x), x=1, x=2, and the x-axis.
Solution: Since the given function is decreasing and lies below the x-axis for x=1 and x=2. Hence the area of the required region will be given by-

\begin{aligned}A=-\int ^{2}_{1}\log _{\dfrac{1}{4}}\left( x\right) dx=-\log _{\dfrac{1}{4}}\left( e\right) \int ^{2}_{1}\ln xdx\\ \Rightarrow -\log _{\dfrac{1}{4}}\left( e\right) \left( x\ln \left( x\right) -x\right) _{1}^{2}\\ \Rightarrow -\log _{\dfrac{1}{4}}\left( e\right) \left( 2\ln \left( 2\right) -1\right) \end{aligned}

Since the area of any region can not be negative, hence the Required Answer is = |A|.

Example: Find the area of the region bounded by the function y = x2-5x+4, x=2, x=3, and the x-axis.
Solution: Since the given function completely lies below the x-axis for x=2 and x=3. Hence the area of the required region will be given by-

\begin{aligned}A=-\int ^{3}_{2}\left( x^{2}-5x+4\right) dx\\ \Rightarrow -\left[ \left( \dfrac{x^{3}}{3}\right) _{2}^{3}-5\left( \dfrac{x^{2}}{2}\right) _{2}^{3}+4\left( x\right) _{2}^{3}\right] \\ \Rightarrow -\dfrac{13}{6}\Rightarrow \left| A\right| =\dfrac{13}{6}\end{aligned}

So this is it from this tutorial. Hoping you people have attempted a few problems based on this tutorial. In the next tutorial, we will discuss the topic “area of the region bounded by the function that changes sign”.

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