How to Find the Area Under a Curve
Area Under the Curve
Area under the curve is calculated by different methods, of which the antiderivative method of finding the area is most popular. The area under the curve can be found by knowing the equation of the curve, the boundaries of the curve, and the axis enclosing the curve. Generally, we have formulas for finding the areas of regular figures such as square, rectangle, quadrilateral, polygon, circle, but there is no defined formula to find the area under the curve. The process of integration helps to solve the equation and find the required area.
For finding the areas of irregular plane surfaces the methods of antiderivatives are very helpful. Here we shall learn how to find the area under the curve with respect to the axis, to find the area between a curve and a line, and to find the area between two curves.
| 1. | How to Find Area Under The Curve? |
| 2. | Different Methods to Find Area Under The Curve |
| 3. | Formula For Area Under The Curve |
| 4. | Area Under The Curve - Circle |
| 5. | Area Under The Curve - Parabola |
| 6. | Area Under The Curve - Ellipse |
| 7. | Area Between a Curve and A-Line |
| 8. | Area Between Two Curves |
| 9. | Solved Examples |
| 10. | Practice Questions |
| 11. | FAQs on Area Under The Curve |
How to Find Area Under The Curve?
The area under the curve can be calculated through three simple steps. First, we need to know the equation of the curve(y = f(x)), the limits across which the area is to be calculated, and the axis enclosing the area. Secondly, we have to find the integration (antiderivative) of the curve. Finally, we need to apply the upper limit and lower limit to the integral answer and take the difference to obtain the area under the curve.
Area = \(_a\int^b y.dx \)
= \(_a\int^b f(x).dx\)
=\( [g(x)]^b_a\)
=\( g(b) - g(a)\)
Different Methods to Find Area Under The Curve
The area under the curve can be computed using three methods. Also, the method used to find the area under the curve depends on the need and the available data inputs, to find the area under the curve. Here we shall look into the below three methods to find the area under the curve.
Method - I: Here the area under the curve is broken down into the smallest possible rectangles. The summation of the area of these rectangles gives the area under the curve. For a curve y = f(x), it is broken into numerous rectangles of width \(\delta x\). Here we limit the number of rectangles up to infinity. The formula for the total area under the curve is A = \(\lim_{x \rightarrow \infty}\sum _{i = 1}^nf(x).\delta x\).
Method - II: This method also uses a similar procedure as the above to find the area under the curve. Here the area under the curve is divided into a few rectangles. Further, the areas of these rectangles are added to get the area under the curve. This method is an easy method to find the area under the curve, but it only provides an approximate value of the area under the curve.
Method - III: This method makes use of the integration process to find the area under the curve. To find the area under the curve by this method integration we need the equation of the curve, the knowledge of the bounding lines or axis, and the boundary limiting points. For a curve having an equation y = f(x), and bounded by the x-axis and with limit values of a and b respectively, the formula for the area under the curve is A = \( _a\int^b f(x).dx\)
Formula For Area Under the Curve
The area of the curve can be calculated with respect to the different axes, as the boundary for the given curve. The area under the curve can be calculated with respect to the x-axis or y-axis. For special cases, the curve is below the axes, and partly below the axes. For all these cases we have the derived formula to find the area under the curve.
Area with respect to the x-axis: Here we shall first look at the area enclosed by the curve y = f(x) and the x-axis. The below figures presents the area enclosed by the curve and the x-axis. The bounding values for the curve with respect to the x-axis are a and b respectively. The formula to find the area under the curve with respect to the x-axis is A = \(_a\int^b f(x).dx\)
Area with respect to the y-axis: The area of the curve bounded by the curve x = f(y), the y-axis, across the lines y = a and y = b is given by the following below expression. Further, the area between the curve and the y-axis can be understood from the below graph.
A = \(_a\int ^bx.dy = _a\int^b f(y).dy\)
Area below the axis: The area of the curve below the axis is a negative value and hence the modulus of the area is taken. The area of the curve y = f(x) below the x-axis and bounded by the x-axis is obtained by taking the limits a and b. The formula for the area above the curve and the x-axis is as follows.
A = |\(_a\int ^bf(x).dx\)|
Area above and below the axis: The area of the curve which is partly below the axis and partly above the axis is divided into two areas and separately calculated. The area under the axis is negative, and hence a modulus of the area is taken. Therefore the overall area is equal to the sum of the two areas(\(A = |A_1 |+ A_2\)).
A = |\(_a\int ^bf(x).dx\)| + \(_b\int ^cf(x).dx\)
Area Under The Curve - Circle
The area of the circle is calculated by first calculating the area of the part of the circle in the first quadrant. Here the equation of the circle x2 + y2 = a2 is changed to an equation of a curve as y = √(a2 - x2). This equation of the curve is used to find the area with respect to the x-axis and the limits from 0 to a.
The area of the circle is four times the area of the quadrant of the circle. The area of the quadrant is calculated by integrating the equation of the curve across the limits in the first quadrant.
A = 4\(\int^a_0 y.dx\)
= 4\(\int^a_0 \sqrt{a^2 - x^2}.dx\)
= 4\([\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}Sin^{-1}\frac{x}{a}]^a_0\)
= 4[((a/2)× 0 + (a2/2)Sin-11) - 0]
= 4(a2/2)(π/2)
= 2πr
Hence the area of the circle is πa2 square units.
Area Under a Curve - Parabola
A parabola has an axis that divides the parabola into two symmetric parts. Here we take a parabola that is symmetric along the x-axis and has an equation y2 = 4ax. This can be transformed as y = √(4ax). We first find the area of the parabola in the first quadrant with respect to the x-axis and along the limits from 0 to a. Here we integrate the equation within the boundary and double it, to obtain the area of the whole parabola. The derivations for the area of the parabola is as follows.
\(\begin{align}A &=2 \int_0^a\sqrt{4ax}.dx\\ &=4\sqrt a \int_0^a\sqrt x.dx\\& =4\sqrt a[\frac{2}{3}.x^{\frac{3}{2}}]_0^a\\&=4\sqrt a ((\frac{2}{3}.a^{\frac{3}{2}}) - 0)\\&=\frac{8a^2}{3}\end{align}\)
Therefore the area under the curve enclosed by the parabola is \(\frac{8a^2}{3}\) square units.
Area Under a Curve - Ellipse
The equation of the ellipse with the major axis of 2a and a minor axis of 2b is x2/a2 + y2/b2 = 1. This equation can be transformed in the form as y = b/a .√(a2 - x2). Here we calculate the area bounded by the ellipse in the first coordinate and with the x-axis, and further multiply it with 4 to obtain the area of the ellipse. The boundary limits taken on the x-axis is from 0 to a. The calculations for the area of the ellipse are as follows.
\(\begin{align}A &=4\int_0^a y.dx \\&=4\int_0^4 \frac{b}{a}.{a^2 - x^2}.dx\\&=\frac{4b}{a}[\frac{x}{2}.\sqrt{a^2 - x^2} + \frac{a^2}{2}Sin^{-1}\frac{x}{a}]_0^a\\&=\frac{4b}{a}[(\frac{a}{2} \times 0) + \frac{a^2}{2}.Sin^{-1}1) - 0]\\&=\frac{4b}{a}.\frac{a^2}{2}.\frac{\pi}{2}\\&=\pi ab\end{align}\)
Therefore the area of the ellipse is πab sq units.
Area Under The Curve - Between a Curve and A-Line
The area between a curve and a line can be conveniently calculated by taking the difference of the areas of one curve and the area under the line. Here the boundary with respect to the axis for both the curve and the line is the same. The below figure shows the curve \(y_1\) = f(x), and the line \(y_2\) = g(x), and the objective is to find the area between the curve and the line. Here we take the integral of the difference of the two curves and apply the boundaries to find the resultant area.
A = \(\int^b_a [f(x) - g(x)].dx\)
Area Under a Curve - Between Two Curves
The area between two curves can be conveniently calculated by taking the difference of the areas of one curve from the area of another curve. Here the boundary with respect to the axis for both the curves is the same. The below figure shows two curves \(y_1\) = f(x), and \(y_2\) = g(x), and the objective is to find the area between these two curves. Here we take the integral of the difference of the two curves and apply the boundaries to find the resultant.
A = \(\int^b_a [f(x) - g(x)].dx\)
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Example 1: Find the area under the curve, for the region bounded by the circle x2 + y2 = 16 in the first quadrant.
Solution:
The given equation of the circle is x2 + y2 = 16
Simplifying this equation we have y = \(\sqrt{4^2 - x^2}\)
A = \(\int^4_0 y.dx\)
= \(\int^4_0 \sqrt{4^2 - x^2}.dx\)
= \([\frac{x}{2}\sqrt{4^2 - x^2} + \frac{4^2}{2}Sin^{-1}\frac{x}{4}]^4_0\)
= [((4/2)× 0 + (16/2)Sin-11) - 0]
= (16/2)(π/2)
= 4π
Answer: Therefore the area of the region bounded by the circle in the first quadrant is 4π sq units -
Example 2: Find the area under the curve, for the region enclosed by the ellipse x2/36 + y2/25 = 1.
Solution:
The given equation of the ellipse is.x2/36 + y2/25 = 1
This can be transformed to obtain y = \(\frac{5}{6}\sqrt{6^2 - x^2}\)
\(\begin{align}A &=4\int_0^6 y.dx \\&=4\int_0^6 \frac{5}{6}.\sqrt{6^2 - x^2}.dx\\&=\frac{20}{6}[\frac{x}{2}.\sqrt{6^2 - x^2} + \frac{6^2}{2}Sin^{-1}\frac{x}{6}]_0^6\\&=\frac{20}{6}[(\frac{6}{2} \times 0) + \frac{6^2}{2}.Sin^{-1}1) - 0]\\&=\frac{20}{6}.\frac{36}{2}.\frac{\pi}{2}\\&=30\pi \end{align}\)
Answer: Therefore the area of the ellipse is 30π sq units.
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FAQs on Area Under The Curve
How to Find the Area Under the Curve?
The area under the curve can be found using the process of integration or antiderivative. For this, we need the equation of the curve(y = f(x)), the axis bounding the curve, and the boundary limits of the curve. With this the area bounded under the curve can be calculated with the formula A = \(_a\int^b y.dx\)
What Are the Different Methods to Find the Area Under the Curve?
There are three broad methods to find the area under the curve. The area under the curve is calculated by dividing the area space into numerous small rectangles, and then the areas are added to obtain the total area. The second method is to divide the area into a few rectangles and then the areas are added to obtain the required area. The third method is to find the area with the help of integration.
What Does Area Under the Curve Mean?
The area under the curve means the area bounded by the curve, the axis, and the boundary points. The area under the curve is a two-dimensional area, which has been calculated with the help of the coordinate axes and by using the integration formula.
What Does Area Under the Curve Represent?
The area under the curve represents the area enclosed under the curve and the axis, which is marked with limiting points. This area under the curve gives the area of the irregular plane shape in a two-dimensional array.
What Is Area Under the Curve in Velocity Time Graph?
In the velocity-time graph, the velocity is graphed with respect to the y-axis, and the time is taken on the x-axis. With this, the area is assumed to be the product of velocity and time and it gives the distance covered. Hence the area under the curve of the velocity-time graph gives the distance covered.
How to Interpret Area Under the Curve?
The area under the curve is the area between the curve and the coordinate axis. Further boundaries are applied across the curve with respect to the axis to obtain the required area. The area under the curve is generally the area of irregular shapes that do not have any area formulas in geometry.
How to Calculate Area Under the Curve Without Integration?
The area under the curve can be calculated even without the use of integration. The area under the curve can be broken into smaller rectangles and then the summation of these areas gives the areas under the curve. Also another method is to break the area under the curve into few rectangles, and then we can take the respective areas to obtain the area under the curve.
How to Approximate the Area Under the Curve?
The area under the curve can be approximately calculated by breaking the area into small parts as small rectangles. And the areas of these rectangles can be calculated and the summation of it gives the area under the curve. Another way to find the approximate area under the curve is to draw a set of few big rectangles and then take a summation of their areas. Further, we can simply find the exact area under the curve with the help of definite integrals.
When to Use Area Under the Curve?
The area under the curve is useful to find the area of irregular shapes in a plane area. We generally find formulas to find the area of a circle, square, rectangle, quadrilaterals, polygon, but we do not have any means to find the area of irregular shapes. Here we use the concept of definite integrals to obtain the area values.
When Is the Area Under the Curve Negative?
The area under the curve is negative if the curve is under the axis or is in the negative quadrants of the coordinate axis. For this also the area of the curve is calculated using the normal method and a modulus is applied to the final answer. Even with the negative answer, only the value of the area is taken, without considering the negative sign of the answer.
How to Find the Area Under a Curve
Source: https://www.cuemath.com/calculus/area-under-the-curve/