An important use of integration is to calculate area between two curves. We already know how to calculate the area under a curve using integration; now let’s see how to find the area between two intersecting curves. It is the area of space that, within the predetermined bounds, lies between two linear or non-linear curves.
Although the area between two curves can also be composite, we can easily determine it using integration by making a few little adjustments to the well-known methods for calculating the area under two curves. Let’s talk about the subject in the next section.
Introduction to the Area between Two Curves
Integral calculus can be used to compute the area between two curves, which is the region between two intersecting curves. When we are aware of the equation for two curves and the locations of their intersections, integration can be utilized to determine the area under the curves. As can be seen in the illustration, there are two functions, f(x) and g(x), and we must determine the area between these two curves, which is indicated by the shaded area. The area of the shaded area can then be simply determined using integration. In the section after this, let’s talk more about how this area was calculated.
What is formula of Area between Two Curves?
When attempting to calculate the approximate area of two curves, it is necessary to first split the region into numerous small rectangular strips that are parallel to the y-axis, ranging from x = a to x = b. These rectangular strips will be “dx” in width and “f(x)-g” in height (x). By utilizing integration inside the boundaries of x = a and x = b, we can now determine the area between these two curves. The area of the small rectangular strip is given by the expression dx(f(x) – g(x)). The following formula can be used if f(x) and g(x) are continuous on [a, b] and g(x) f(x) for every x in [a, b].
A = ∫^b_a [ f(x)−g(x)]dx
Area Between Two Curves With Respect to Y
The area between two curves with respect to the y-axis is the method of calculating the areas of the curves whose equation is given in terms of y. Calculating the area along the y-axis is easier compared to calculating the area along the x-axis. In this method, we divide the given region into horizontal strips between the given limits, and by using integration, we add the areas of the horizontal strips to find the area of the section between two curves. If f(y) and g(y) are continuous on [c, d] and g(y) < f(y) for all y in [c, d], then
Area = ∫ba[f(y)−g(y)]dy
Area Between Two Compound Curves
Calculating areas between two compound curve which intersect with each other using above stated formulas will give the incorrect result and curves change places after the intersection. For the curves shown in the image, we divided the intervals into various portions and then calculate individual areas between the curves in each section. Let f(x) and g(x) be continuous in [a,b] interval, the area between the curves will be:
Area = ∫ca|f(x)−g(x)|dx∫ac|f(x)−g(x)|dx
As we see in the region [a, b], f(x) ≥ g(x) and in the region [c, d] g(x) ≥ f(x), so we break the limits into two parts as:
Area = ∫ba(f(x)−g(x))dx+∫cb(g(x)−f(x))dx
What is mean by Area under the Curve?
The region enclosed by the curve, the axis, and the boundary points is referred to as the “area under the curve.” Using the coordinate axes and the integration formula, it is possible to get the two-dimensional area under the curve.
What is Represent Area under the Curve?
The region enclosed by the curve and the axis, which is indicated by limiting points, is represented by the region beneath the curve. The area of the asymmetric plane shape in a two-dimensional array is provided by the region beneath the curve.
How to determine Area under the Curve?
The integration or ant derivative processes can be used to determine the curve’s area under it. For this, we require the curve’s equation (y = f(x)), the curve’s axis boundary, and the curve’s border limitations. This allows for the calculation of the area bounded under the curve using the a = ∫ba y.dx∫aby.dx.