Area between two curves - along x-axis and y-axis

YOU WILL LEARN ABOUT:

How to determine the area between two curves by integrating with respect to $x$ or $y$ , which is a generalization of the concept of finding the area between a curve and the $x$ -axis $(y = 0)$ .

The area between the continuous curves

$y = f_{1} (x)$ and $y = f_{2} (x)$ and between $x = a$ and $x = b$ is given by $\int_{a}^{b} [f_{1} (x) - f_{2} (x)] d x$ , where $f_{1} (x) = f_{2} (x)$ for all $x$ between $a$ and $b$ .
$x = g_{1} (y)$ and $x = g_{2} (y)$ and between $y = c$ and $y = d$ is given by $\int_{d}^{c} [g_{1} (y) - g_{2} (y)] d y$ , where $g_{1} (y) = g_{2} (y)$ for all $y$ between $c$ and $d$ .

Area calculations are important in many applications. The area of many regions can be determined using integrals by thinking of the regions as lying between two curves. The following are examples of such regions.

The area between the marginal revenue and marginal cost functions, from x = a to
x = b, for a manufacturer represents the increase in profits as the production level increases from a to b units.

Medical professionals use a quantity known as quality-adjusted life years (QALYs) as a measure of health outcomes which incorporates both the quality and quantity of life lived. When considering some sort of medical intervention, the area between two QALY vs. time curves represents the gain in quality-adjusted life years by the intervention.

An environmental contractor hired to clean up an oil spill needs to determine the area of an oil slick, which can be modeled as the region between two curves, to be able to determine the proper amount of dispersants and biodegradation agents to use to clean the spill.

Click on the tabs on the right to watch videos on rotating a region about the x-axis and y-axis.

X Axis

Y Axis

Difference =

Number of rectangles:4
Note:The first rectangle has a height of zero.

Integrate with respect to x-axis y-axis

Function 1

Function 2

Function 3

Function 4