Calculus Basics: December 2013

Bounded Areas using Integration
The Integral $\int_{x=a}^{x=b}ydx$ represents area bounded by the curve y=f(x) on one side, the line x=a and x=b on other two sides, x-axis on the fourth side.

If the area lies below x-axis $\int_{x=a}^{x=b}ydx$ gives negative value as area. Therefore the formula in such case is, Area= $\left |\int_{x=a}^{x=b}ydx \right |$ or $\int_{x=a}^{x=b} (-y)dx$

The Integral $\int_{y=c}^{y=d} xdy$ represents area bounded by the curve x=f(y) on one side, the line y=c and y=d on other two sides, y-axis on the fourth side.

If the area lies left of y-axis $\int_{y=c}^{y=d} xdy$ gives negative value as area. Therefore the formula in such

case is, Area= $\left |\int_{y=c}^{y=d} xdy \right |$ or $\int_{y=c}^{y=d} (-x)dy$

When using Integrals to find the Area we always get the area between the curve and the axis (dx or dy). For example, (refer the figure below) while finding area bounded by a parabola open upwards from x=a to x=b and x-axis we will not get area A1 but we arrive at area A2.

If the Area under consideration lies above as well as below x-axis (see figure below), then you have to find area like this, $\int_{a}^{b} ydx =\int_{a}^{c}ydx + \int_{c}^{d}(-y)dx + \int_{d}^{b}ydx$ or

$\int_{a}^{b} ydx =\int_{a}^{c}ydx + \left |\int_{c}^{d}ydx \right | + \int_{d}^{b}ydx$

Calculus Basics

maths operators

Saturday, 7 December 2013

Areas using Integrals