## Applications of the Integrals

# Area under Simple Curves

- The area region bounded by the y = f(x), the x - axis and the lines x = a, x = b is =\begin{vmatrix}\int_{a}^{b} y \ dx \end{vmatrix}=\begin{vmatrix}\int_{a}^{b} f(x) \ dx \end{vmatrix}
- The area region bounded by the curve y = f(x) the y-axis and the lines y = c, y = d is =\begin{vmatrix}\int_{c}^{d} x \ dy \end{vmatrix}=\begin{vmatrix}\int_{c}^{d} f^{-1}(y) \ dy \end{vmatrix}
- The area region bounded by y = f(x), y = g(x) the lines x = a, x = b is \begin{vmatrix}\int_{a}^{b} f(x)-g(x) \ dx \end{vmatrix}
- The area region bounded by y = f(x) and y = g(x) and the lines y = c, y = d is \begin{vmatrix}\int_{c}^{d} f^{-1}(y)-g^{-1}(y) \ dy \end{vmatrix}
- If y = f(x) and y = g(x) and these curves intersect at P(α
_{1}, β_{1}), Q (α_{2}, β_{2}), then the area between curves is \begin{vmatrix}\int_{\alpha_{1}}^{\alpha_{2}} f(x)-g(x) \ dx \end{vmatrix} on x-axis \begin{vmatrix}\int_{\beta_{1}}^{\beta_{2}} f^{-1}(y)-g^{-1}(y) \ dy\end{vmatrix} on y-axis - The area of the ellipse \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 is πab sq.units
- The area of the circle x
^{2}+ y^{2}= a^{2}is π a^{2}sq.units. - The area region bounded by the curve y = sin ax (or) cos ax and x-axis is \frac{2}{a} sq.units by one arc.
- The area region bounded by the curve y = sin ax (or) y = cos ax and x-axis in [0, nπ] is \frac{2n}{a} sq.units

### Part1: View the Topic in this video From 14:04 To 48:16

### Part2: View the Topic in this video From 04:50 To 12:40

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1. The area of the region bounded by the curve y = f(x), x-axis and the lines x = a and x = b (b > a) is given by the formula: Area = \int_{a}^{b} y \ dx=\int_{a}^{b} f(x)\ dx.

2. The area of the region bounded by the curve x = Φ(y), y-axis and the lines y = c, y = d is given by the formula: Area = \int_{c}^{d} x \ dy=\int_{c}^{d} \phi(y)\ dy.