Area bounded by a single curve y = f(x)
Area Bounded by a Single Curve y = f(x)
When we talk about the area bounded by a single curve, we are referring to the region enclosed by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b, where a and b are the bounds of the interval we are interested in.
Calculating the Area
To calculate the area under the curve y = f(x) from x = a to x = b, we use the definite integral. The area A is given by:
$$ A = \int_{a}^{b} f(x) \, dx $$
This integral sums up an infinite number of infinitesimally small rectangles under the curve from x = a to x = b. If the curve lies above the x-axis in this interval, the area is simply the integral. If the curve crosses the x-axis, the area is calculated as the sum of the absolute values of the integrals of the segments above and below the x-axis.
Important Points
| Point | Description |
|---|---|
| Positive Area | If f(x) is positive over [a, b], then the area is simply the integral of f(x) from a to b. |
| Negative Area | If f(x) is negative over [a, b], the area is the integral of the absolute value of f(x), which is equivalent to the negative of the integral from a to b. |
| Curve Crossing X-axis | If f(x) crosses the x-axis, the area is calculated by breaking the integral into sections where f(x) is positive or negative and summing their absolute values. |
| Units | The area is always a positive quantity and has units that are the square of the units on the x-axis. |
Formulas
- Area under the curve above the x-axis: $ A = \int_{a}^{b} f(x) \, dx $
- Area under the curve below the x-axis: $ A = -\int_{a}^{b} f(x) \, dx $
- Total area when the curve crosses the x-axis: $ A = \int_{a}^{c} |f(x)| \, dx + \int_{c}^{b} |f(x)| \, dx $ where c is the x-coordinate of the point where f(x) crosses the x-axis.
Examples
Example 1: Area under a Positive Curve
Consider the curve y = x^2 over the interval [0, 2]. The area under the curve is calculated as:
$$ A = \int_{0}^{2} x^2 \, dx = \left[\frac{x^3}{3}\right]_{0}^{2} = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3} $$
Example 2: Area under a Negative Curve
Consider the curve y = -x over the interval [1, 3]. The area under the curve is:
$$ A = -\int_{1}^{3} (-x) \, dx = \int_{1}^{3} x \, dx = \left[\frac{x^2}{2}\right]_{1}^{3} = \frac{3^2}{2} - \frac{1^2}{2} = \frac{9}{2} - \frac{1}{2} = 4 $$
Example 3: Area with a Curve Crossing the X-axis
Consider the curve y = x^2 - 4 over the interval [-3, 3]. The curve crosses the x-axis at x = -2 and x = 2. The area is calculated as:
$$ A = \int_{-3}^{-2} |x^2 - 4| \, dx + \int_{-2}^{2} |x^2 - 4| \, dx + \int_{2}^{3} |x^2 - 4| \, dx $$
This requires evaluating three separate integrals and summing their absolute values to find the total area.
Conclusion
The area bounded by a single curve y = f(x) is a fundamental concept in calculus. It is important to understand how to set up and evaluate definite integrals to find the area under or between curves. Remember that the area is always a positive quantity, and when dealing with curves that cross the x-axis, you must consider the absolute value of the integrals of the segments.