Area under the Curve Formula
The area under a curve between two points is found out by doing a definite integral between the two points. To find the area under the curve y = f(x) between x = a & x = b, integrate y = f(x) between the limits of a and b. This area can be calculated using integration with given limits.
Formula to Calculate the Area Under a Curve
The formula for Area under the Curve = ∫ab f(x)dx
Solved Example
Question : Calculate the area under the curve of a function, f(x) = 7 – x2, the limit is given as x = -1 to 2.
Solution:
Given function is, f(x) = 7- x2 and limit is x = -1 to 2
\(\begin{array}{l}\large Area = \int_{-1}^{2}(7-x^{2})dx\end{array} \)
\(\begin{array}{l}\large = \left ( 7x-\frac{1}{3}x^{3}\right)|_{-1}^{2}\end{array} \)
\(\begin{array}{l}\large = \left [ 7.2-\frac{1}{3}(8) \right ]-\left [ 7(-1)-\frac{1}{3}(-1)\right ]\end{array} \)
= [(42 – 8)/3] – [(1 – 21)/3]
= (34 + 20)/3
= 54/3
= 18 sq.units
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