Finding the Area between Two Curves
The easiest way to think about the area between two curves: the area between the curves is the area below the upper curve minus the area underneath the lower curve. You can figure out the area between two curves by calculating the difference between the definite integrals of two functions. In 2-D geometry, the area is a volume that describes the region occupied by the two-dimensional figure. Two functions are needed to determine the area, say f(x) and g(x), and the integral limits from 'a’ to ‘b’ (b should be >a) of the function, that acts as the bespoke of the curve.
Formula to Find the Area between Two Curves
The basic mathematical expression written to compute the area between two curves is as follows:
If P: y = f(x) and Q : y = g(x) and x1 and x2 are the two limits,
Now the standard formula of- Area Between Two Curves, A=∫x2x1[f(x)−g(x)]
Through this topic, you should be able to:
ü find the area between two curves
ü find the area between two curves by integration
Calculating Areas Between Two Curves by Integration
1. Area under a curve – Region encircled by the given function, vertical lines and the x –axis.
2. Area Under a Curve – region encircled by the given function, horizontal lines and the y –axis.
3. Area between curves expressed by given two functions.
In case f(x) is a nonnegative and continuous function of x on the closed interval [a, b], then the area of the region enclosed by the graph of ‘f’, the x-axis and the vertical lines x=a and x=b is given by:
b
a
Area f (x)dx
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When computing the area under a curve f(x), follow the below set of instructions:
Shade the area.
Identify the boundaries a and b,
Establish the definite integral,
Integrate.
Calculating Areas Between Curves Using Double Integrals
The common application of the single variable integral is to compute the area under a curve f(x) over some interval [a,b] by integrating f(x) over that interval. That being said, you can sometimes also apply double integrals to compute areas between curves. However, the proposition is not the same. It's fairly simple to understand the tactic to achieve this once you can envision how to use a single integral to find the length of the interval.
Now you must be thinking as to What happens if you integrate the function f(x)=1 over the interval [a,b]? You can compute that
∫baf(x)dx=∫ba1dx=x∣∣ba=b−a.
The integral of the function f(x) =1 is merely the length of the interval [a,b]. Fact is that it also comes about as the area of the rectangle of height 1 and length (b−a), but we can explain it as the length of the interval [a,b].
You can apply the similar trick for finding areas with double integrals. The integral of a function f(x,y) over a region D can be simplified as the quantity beneath the surface z=f(x,y) over the region D. As executed above, we can attempt the tactic of integrating the function f(x,y)=1 over the region D. This would give the volume under the function f(x,y)=1 over D. But the integral of f(x,y)=1 is also the area of the region D. This can be a nifty way of calculating the area of the region D. Hence, if we If we entitle ‘A’ be the area of the region D, we can write it in the form of :-
A=∬DdA.
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Solved Example
As we said above, practice is the key to master over calculating area curves. So let’s begin with some fun exercises.
Problem
Find the area encircled by the following curves: 4,= 0, x = y - x =y
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Solution
Determining the boundaries: y = x² - 4, y=0 which implies x²- 4=0, therefore, (x-2) (x+2) = 0 x = - 2 or x = 2.
With the preview of the graph we can observe that 2=x is the boundary at ‘a’. The assessment of 2- =x is long away from encircling the area of the region. This is why the graph here plays a crucial part in helping identify the appropriate outcome to the problem. The value of the other boundary is provided by the equation of the vertical line 4,=x .
Boundaries are: 2,=a and 4,=b
Now, Establish the integral:
A= ò ò A f (x)dx (x 4)dx
Solving,
ò (x² - 4) dx= \[(\frac{1}{3} x^{3} - 4x)\] ò = (\[\frac{1}{3}.(4)^{3}) - 4.4) - (\frac{1}{3}.{2}^{3} - 4.2)\]
= \[(\frac{64}{3} - 16) - (\frac{8}{3} - 8) = \frac{64}{3} - 16 - \frac{8}{3} + 8 - \frac{56}{3} - 8 - \frac{32}{3}\]
Thus, the area encircled by the curves y - x² -4, y=0, x-4 = 32ö ç3 square units.
Fun Facts
Multiple integrals are much easier to use than single integrals’ in finding area with integrals
Drawing the sketch or graph beforehand makes it easy to find areas of the region that should be subtracted.
It may be a requisite to find the areas of curves in several parts and add up the outcomes to achieve the final result.
FAQs on Area Between Two Curves Calculus
1. What is the general formula to find the area between two curves in calculus?
The general formula to find the area between two continuous functions, y = f(x) and y = g(x), from x = a to x = b, where f(x) ≥ g(x) on the interval [a, b], is given by the definite integral: Area = ∫ₐᵇ [f(x) - g(x)] dx. In this formula, f(x) represents the upper curve and g(x) represents the lower curve.
2. What are the key steps to calculate the area between two curves, y = f(x) and y = g(x)?
To calculate the area between two curves as per the CBSE Class 12 Maths syllabus, you should follow these steps:
- Find Points of Intersection: Set the two functions equal to each other (f(x) = g(x)) to find the x-values where the curves intersect. These values often define the limits of integration, 'a' and 'b'.
- Identify the Upper and Lower Curve: In the interval [a, b], determine which function has the greater value. The function with the larger value is the upper curve, f(x), and the other is the lower curve, g(x).
- Set Up the Definite Integral: Write the integral using the formula: Area = ∫ₐᵇ [Upper Curve - Lower Curve] dx.
- Integrate and Evaluate: Calculate the definite integral to find the numerical value of the area.
3. When should you calculate the area between two curves with respect to the y-axis instead of the x-axis?
You should calculate the area with respect to the y-axis when the curves are more easily expressed as functions of y, i.e., in the form x = f(y) and x = g(y). This method is particularly useful when integrating with respect to x would require splitting the area into multiple regions. In this case, the formula becomes Area = ∫cᵈ [Right Curve - Left Curve] dy, where 'c' and 'd' are the y-values of the integration limits.
4. Why is definite integration the fundamental method for calculating the area between curves?
Definite integration is used because it fundamentally represents the summation of infinitesimally small areas. To find the area between two curves, we imagine slicing the region into an infinite number of thin vertical rectangles. The height of each rectangle is the difference between the upper and lower curves, [f(x) - g(x)], and its width is an infinitesimally small change in x, denoted as 'dx'. The integral, ∫, is the symbol for summing up the areas of all these rectangles from the starting point 'a' to the endpoint 'b'.
5. How does the Fundamental Theorem of Calculus justify the formula for the area between two curves?
The Fundamental Theorem of Calculus connects differentiation and integration. The formula for the area between curves, Area = ∫ₐᵇ [f(x) - g(x)] dx, relies on this theorem. The integral of the rate of change of an area gives the total change in area. Here, the function [f(x) - g(x)] represents the 'height' of the area at any point x. By integrating this height function, we are essentially accumulating the area under f(x) and subtracting the area under g(x), which the Fundamental Theorem allows us to calculate precisely by evaluating the antiderivative at the endpoints 'a' and 'b'.
6. What happens if the two curves intersect multiple times within the interval of integration?
If two curves intersect multiple times, you must break the area calculation into separate integrals for each sub-interval where one curve is consistently above the other. For example, if f(x) is above g(x) on [a, b] but g(x) is above f(x) on [b, c], the total area would be the sum of two integrals: Total Area = ∫ₐᵇ [f(x) - g(x)] dx + ∫bᶜ [g(x) - f(x)] dx. You cannot use a single integral from 'a' to 'c' as the parts where the lower curve is subtracted from the upper would cancel out.
7. Can the calculated area between two curves be negative? What does a negative result signify?
The area itself, being a physical quantity, cannot be negative. However, the result of the definite integral ∫ₐᵇ [f(x) - g(x)] dx can be negative. A negative result signifies that you have incorrectly identified the upper and lower curves. It means that over the interval of integration, g(x) was actually the upper curve and f(x) was the lower. The magnitude of the result is still the correct area. To fix this, you can simply take the absolute value of the result or ensure the integrand is always [Upper Curve - Lower Curve].
8. What is a common mistake students make when setting up the integral for the area between two curves?
A very common mistake is incorrectly setting up the integrand by not properly identifying the upper and lower functions within the specified interval. Students sometimes subtract the functions in a fixed order without checking which one has a greater y-value for a given x in the interval. This leads to a negative result for the integral, which, if not interpreted correctly, can lead to an incorrect final answer. Always test a point within the interval or sketch the graph to confirm which function is on top.
