Area Under the Curve Formula
The area under the curve formula can be determined by performing a definite integral between the given limits. Area under a curve y=f(x) can be integrating the function between x=a and x=b. For calculating the area under the curve we divide the whole area in the form of few rectangular strips of height/length = f(x0) and breadth = dx and the total area under the curve can be approximately obtained by adding the areas of all the rectangular strips. Now the area under a curve formula can be calculated by using integration with given limits. Let us understand the area under the curve formula in detail using solved examples in the following section.
What is Area Under The Curve Formula?
The area under a curve can be determined by performing a definite integral between the given limits. The area under a curve y=f(x) can be integrating the function between x=a and x=b and the formula for the area under a curve is given by:
Area Under The Curve =\(\int_{a}^b f(x)dx\)
where,
- a and b are the limits of integration
- f(x) is the function
Let's take a quick look at a couple of examples to understand the area under the curve formula, better.
Solved Examples Using Area Under The Curve Formula
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Example 1: Find the area under the curve y=x2+2 between x = 0 and x = 4.
Solution:
To find: Area under the curve.y=f(x)=x2+2, a = 0 and b = 4 (given)
Using area under the curve formula,
Area under the curve =\(\int_{a}^b f(x)dx\)
Putting the values, we get,
Area =\(\int_{0}^4(x^2 + 2)dx\)
= \(\left[\dfrac{1}{3}x^3 + 2x \right]_{0}^{4}\)
= \(\left [\dfrac{1}{3}4^3 + 2(4) \right] - \left [\dfrac{1}{3}(0)^3 + 2(0) \right]\)= 21.33 + 8
= 29.33
Answer: Area under the curve is 29.33 sq units.
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Example 2: Find the area under the curve y = 3x between x = 1 and x = 5.
Solution:
To find: Area under the curve.
y = f(x) = 3x, a = 1 and, b = 5 (given)
Using area under the curve,
Area under the curve =\(\int_{a}^b f(x)dx\)
Putting the values, we get,
Area Under The Curve =\(\int_{5}^1 f(3x)dx\)
\(\left [\dfrac{3}{2}x^2 \right]_{1}^{5}\)
=\(\left [\dfrac{3}{2}(5)^2\right] - \left [\dfrac{1}{3}(1)^2 \right]\)
=37.166
Answer: Area under the curve is 37.166 sq units.
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