how to find area under a curve
In Calculus, we accept seen definite integral as a limit of a sum and we know how to evaluate it using the fundamental theorem of calculus. In the upcoming discussion, we will run into an easier style of finding the area divisional past any curve and x-axis betwixt given coordinates. To determine the area under the curve nosotros follow the beneath method.
How to Decide the Expanse Under the Curve?
Let us assume the curve y=f(x) and its ordinates at the x-axis exist x=a and x=b. Now, we demand to evaluate the expanse bounded by the given bend and the ordinates given by x=a and x=b.
The area under the curve tin be assumed to be made upwards of many vertical, extremely thin strips. Permit us take a random strip of height y and width dx equally shown in the figure given above whose area is given past dA.
The expanse dA of the strip can be given every bit y dx. Also, nosotros know that any point of the curve, y is represented as f(x). This surface area of the strip is called an elementary area. This strip is located somewhere between x=a and ten=b, between the ten-centrality and the curve. At present, if nosotros demand to find the total area bounded by the curve and the ten-axis, between x=a and x=b, then it tin be considered to exist made of an infinite number of such strips, starting from x=a to x=b. In other words, adding the simple areas betwixt the thin strips in the region PQRSP volition give the total expanse.
Mathematically, it can be represented as:
\(\brainstorm{assortment}{l} A = \int\limits_{a}^b dA = \int\limits_{a}^b ydx = \int\limits_{a}^b f(ten)dx\end{array} \)
Using the aforementioned logic, if nosotros desire to summate the area under the bend x=yard(y), y-axis between the lines y=c and y=d, it will be given past:
\(\begin{assortment}{l} A = \int\limits_{c}^d xdy = \int\limits_{c}^d g(y)dy\stop{array} \)
In this example, we need to consider horizontal strips as shown in the effigy above.
Also, note that if the curve lies below the 10-centrality, i.eastward. f(x) <0 and then following the same steps, you will get the area under the curve and 10-centrality between x=a and x=b as a negative value. In such cases, take the absolute value of the area, without the sign, i.due east.|
\(\begin{array}{l}\int\limits_{a}^b f(x)dx|\end{array} \)
Some other possibility is that, when some portion of the bend may lie to a higher place the x-centrality and some portion below the x-axis, as shown in the figure,
Hither A1<0 and A2>0. Hence, this is the combination of the beginning and second case. Hence, the total area will exist given as |Ai|+Atwo
Expanse Under the Curve Example
Let us consider an case, to understand the concept in a better way.
We demand to discover the total expanse enclosed past the circle x2+y2=ane
Expanse enclosed by the whole circumvolve = 4 x area enclosed OABO
=4
\(\begin{array}{l}\int\limits_{0}^one ydx \end{array} \)
(considering vertical strips)\(\begin{array}{l} =4 \int\limits_{0}^ane \sqrt{{ane}-{x}^{2}}\stop{array} \)
On integrating, we get,
\(\brainstorm{array}{l}=4 [\frac{x}{two}\sqrt{{1}-{x}^{2}}+ \frac{one}{2} {sin}^{-1}x{]}_{0}^{1}\cease{array} \)
= 4 × ane/2 10 π/2
= π
So the required area is π square units.
To learn more than about the area under the bend download BYJU'Southward – The Learning App.
Source: https://byjus.com/maths/area-under-curve-calculus/
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