33. Applications of Definite Integrals1
Dated: 30-10-2024
Area between 2 Curves
Let there be 2 functions2 \(f(x)\) and \(g(x)\) defined over the interval3 \([a, b]\) such that \(f(x) \ge g(x)\).
Let \(A_1\) be area under \(f(x)\) and \(A_2\) be area under \(g(x)\) then \(A\), being the area between them can be found out by
\[A = A_1 - A_2\]
\[A=\int_{a}^{b}f(x)dx-\int_{a}^{b}g(x)dx=\int_{a}^{b}(f(x)-g(x))dx\]
It is possible that one of the functions2 is under the x axis. Then in that case, we translate both functions2 above the x axis by some constant value \(m\).
\[A=\int_{a}^{b}(f(x)+m)dx-\int_{a}^{b}(g(x)+m)dx=\int_{a}^{b}(f(x)-g(x))dx\]
This definition also applies if we replace \(x\) with \(y\).
References
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