Area Under the Curve :
Area Under the Curve, also known as the definite integral, is a mathematical concept used to calculate the area enclosed by a curve on a graph. It is a fundamental concept in calculus and is used in a wide range of applications, including physics, engineering, and economics.
To better understand the concept of area under the curve, let’s consider a simple example of a straight line on a graph. Suppose we have a line on a graph with coordinates (x, y), where x is the horizontal axis and y is the vertical axis. The line has a slope of 1, meaning that for every unit increase in the value of x, the value of y increases by 1.
The area under the line can be calculated by dividing the line into small segments and calculating the area of each segment using the formula for the area of a rectangle (length x width). The sum of the areas of all the rectangles would then give us the total area under the line.
Now, let’s consider a more complex example of a curve on a graph. Suppose we have a curve with the equation y = x^2. The curve has a parabolic shape and is defined by the coordinates (x, x^2). To calculate the area under the curve, we can again divide the curve into small segments and calculate the area of each segment using the formula for the area of a rectangle.
However, in this case, the width of the rectangles would be different for each segment, as the curve is not a straight line. To calculate the width of each rectangle, we need to determine the length of the segment and the slope of the curve at each point. The slope of the curve can be calculated using the derivative of the curve’s equation.
Once we have determined the width of each rectangle, we can calculate the area of each segment using the formula for the area of a rectangle. The sum of the areas of all the rectangles would then give us the total area under the curve.
In practice, calculating the area under a curve using this method can be quite tedious and time-consuming, especially for complex curves with many segments. Fortunately, calculus provides a more efficient way to calculate the area under a curve using the concept of the definite integral.
The definite integral is defined as the limit of the sum of the areas of the rectangles as the number of segments approaches infinity. In other words, it is the sum of the areas of an infinite number of infinitesimally small rectangles under the curve.
To calculate the definite integral, we first need to determine the bounds of the integral, which are the lower and upper limits of the curve. In the example above, the bounds would be the minimum and maximum values of x.
Next, we need to determine the function of the curve, which is the equation that defines the curve. In the example above, the function of the curve would be y = x^2.
Finally, we can calculate the definite integral using the following formula:
∫a^b f(x) dx = lim n→∞ Σ (width of each rectangle) x (height of each rectangle)
In this formula, a and b are the bounds of the integral, f(x) is the function of the curve, and dx is a small change in the value of x.
Using this formula, we can calculate the definite integral for the curve in the example above as follows:
∫0^2 x^2 dx = lim n→∞ Σ (width of each rectangle) x (height of each rectangle)