Finding Unknown Parameters Using Area Under Curves

Finding Unknown Parameters Using Area Under Curves

Since we already know that, the area of the region bounded by a function f(x), x-axis, and ordinates x = a and x = b can be given by the equation Area\left( A\right) =\int ^{b}_{a}f\left( x\right) dx

We can observe from the above equation that there is a relation between four items (f(x), A, a, and b). It may also happen that there may be any unknown parameter(s) involved in these four items and we have to find out the value of that parameter or the relationship between those parameters.

We have already covered so many aspects of evaluating area under curves. So in order to understand today’s topic, we will be taking the help of a few examples. After all, we just have to find the value of the parameter or relationship between those parameters. So whatever we learned so far in the topic “Area Under Curves” will be quite enough to do so.

Example: the area of the region enclosed by the curves y = 1/x, y = x2 and x = 1/2 is ln(2) – p, then the value of ‘p’ will be…
Solution: Since the curves y = 1/x, y = x2 intersect each other at (1, 1). Hence the required area will be

\begin{aligned}Area\left( A\right) =\int ^{1}_{1/2}\left( \dfrac{1}{x}-x^{2}\right) dx=\left( \ln x-\dfrac{x^{3}}{3}\right) _{1/2}^{1}\\ \Rightarrow \ln 2-\dfrac{7}{24}\\ \Rightarrow \ln 2-p=\ln 2-\dfrac{7}{24}\\ \Rightarrow p=\dfrac{7}{24}\end{aligned}

Example: Let a curve y = ax1/2+bx passes through a point (1, 2). If the area of the region bounder between the curve, the x-axis and the straight line x = 4 is 8, Then what will be the values of parameters ‘a’ and ‘b’.
Solution: Since the curve y = ax1/2+bx passes through a point (1, 2). Hence we have a + b = 2. The given curve also passes through origin => x ≥ 0. So

\begin{aligned}Area=\int ^{4}_{0}\left( a\sqrt{x}+bx\right) dx=8\\ \Rightarrow \left( \dfrac{2a}{3}x^{3/2}\right) _{0}^{4}+\left( \dfrac{bx^{2}}{2}\right) _{0}^{4}=\dfrac{16a}{3}+8b\\ \Rightarrow \dfrac{16a}{3}+8b=8,a+b=2\\ \Rightarrow a=3,b=-1\end{aligned}

Now attempt at least five questions based on this topic. So this is it from this tutorial. Hoping you people will attempt a few problems based on the topic discussed in this tutorial. In the next tutorial, we will discuss the topic “Area Of The Region Bounded By Shifted Curves”.

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