Area Under The Curve Units
Expanse Under the Bend
Area nether the curve is calculated past unlike methods, of which the antiderivative method of finding the area is nearly pop. The expanse under the curve can exist constitute by knowing the equation of the bend, the boundaries of the curve, and the axis enclosing the curve. Generally, we have formulas for finding the areas of regular figures such as foursquare, rectangle, quadrilateral, polygon, circle, but there is no divers formula to detect the area nether the curve. The process of integration helps to solve the equation and notice the required surface area.
For finding the areas of irregular aeroplane 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 betwixt a curve and a line, and to observe the surface area between two curves.
| 1. | How to Find Area Under The Curve? |
| ii. | Different Methods to Find Area Under The Bend |
| 3. | Formula For Area Under The Curve |
| 4. | Area Nether The Bend - Circle |
| 5. | Area Nether The Curve - Parabola |
| vi. | Area Under The Bend - Ellipse |
| vii. | Surface area Between a Curve and A-Line |
| 8. | Area Between 2 Curves |
| nine. | Solved Examples |
| 10. | Exercise Questions |
| eleven. | FAQs on Area Under The Curve |
How to Find Area Under The Bend?
The area under the curve can be calculated through three simple steps. First, we need to know the equation of the bend(y = f(x)), the limits across which the area is to be calculated, and the axis enclosing the area. Secondly, nosotros take to detect the integration (antiderivative) of the curve. Finally, we demand to employ the upper limit and lower limit to the integral respond and take the divergence to obtain the area nether the curve.
Area = \(_a\int^b y.dx \)
= \(_a\int^b f(x).dx\)
=\( [g(x)]^b_a\)
=\( g(b) - m(a)\)
Unlike Methods to Discover Surface 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 demand and the available data inputs, to find the surface area under the curve. Hither we shall look into the beneath iii methods to find the area under the bend.
Method - I: Hither the area under the curve is broken down into the smallest possible rectangles. The summation of the area of these rectangles gives the surface area nether the curve. For a curve y = f(x), information technology is broken into numerous rectangles of width \(\delta x\). Hither we limit the number of rectangles upwards to infinity. The formula for the total area nether the curve is A = \(\lim_{x \rightarrow \infty}\sum _{i = one}^nf(ten).\delta ten\).
Method - Two: 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. Farther, the areas of these rectangles are added to get the surface area under the curve. This method is an easy method to find the area nether the bend, but it only provides an approximate value of the expanse under the curve.
Method - III: This method makes use of the integration process to find the area under the curve. To find the surface area under the curve past this method integration we need the equation of the curve, the cognition of the bounding lines or centrality, and the boundary limiting points. For a curve having an equation y = f(ten), and bounded by the 10-axis and with limit values of a and b respectively, the formula for the area nether the curve is A = \( _a\int^b f(x).dx\)
Formula For Area Nether the Curve
The area of the curve can exist calculated with respect to the different axes, as the boundary for the given curve. The area under the curve tin can exist calculated with respect to the x-centrality 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 observe the surface area under the bend.
Area with respect to the x-axis: Here we shall beginning await at the area enclosed past the curve y = f(x) and the 10-axis. The below figures presents the expanse enclosed by the curve and the x-axis. The bounding values for the bend with respect to the 10-axis are a and b respectively. The formula to discover the area nether 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 10 = f(y), the y-axis, beyond the lines y = a and y = b is given by the following below expression. Further, the surface area between the bend and the y-centrality can be understood from the beneath graph.
A = \(_a\int ^bx.dy = _a\int^b f(y).dy\)
Expanse below the axis: The area of the bend beneath the centrality is a negative value and hence the modulus of the area is taken. The expanse of the bend y = f(x) below the x-axis and divisional past the 10-axis is obtained by taking the limits a and b. The formula for the area to a higher place the bend and the x-axis is as follows.
A = |\(_a\int ^bf(x).dx\)|
Surface area above and below the axis: The area of the bend which is partly beneath the axis and partly above the axis is divided into two areas and separately calculated. The surface area under the axis is negative, and hence a modulus of the surface area is taken. Therefore the overall area is equal to the sum of the 2 areas(\(A = |A_1 |+ A_2\)).
A = |\(_a\int ^bf(x).dx\)| + \(_b\int ^cf(10).dx\)
Expanse Nether The Curve - Circle
The area of the circle is calculated past first calculating the expanse of the function of the circumvolve in the start quadrant. Here the equation of the circle tenii + y2 = a2 is changed to an equation of a bend as y = √(a2 - 102). This equation of the curve is used to find the surface area with respect to the ten-axis and the limits from 0 to a.
The surface area of the circumvolve is 4 times the area of the quadrant of the circle. The surface area of the quadrant is calculated by integrating the equation of the bend across the limits in the first quadrant.
A = 4\(\int^a_0 y.dx\)
= four\(\int^a_0 \sqrt{a^two - 10^2}.dx\)
= 4\([\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}Sin^{-1}\frac{x}{a}]^a_0\)
= iv[((a/two)× 0 + (a2/2)Sin-11) - 0]
= 4(atwo/2)(π/2)
= 2πr
Hence the area of the circle is πaii square units.
Area Under a Curve - Parabola
A parabola has an centrality that divides the parabola into 2 symmetric parts. Here we have a parabola that is symmetric along the x-centrality and has an equation y2 = 4ax. This can be transformed as y = √(4ax). We outset find the expanse of the parabola in the starting time quadrant with respect to the ten-axis and forth the limits from 0 to a. Here we integrate the equation within the boundary and double information technology, to obtain the area of the whole parabola. The derivations for the area of the parabola is equally follows.
\(\begin{marshal}A &=2 \int_0^a\sqrt{4ax}.dx\\ &=four\sqrt a \int_0^a\sqrt x.dx\\& =4\sqrt a[\frac{2}{3}.x^{\frac{3}{2}}]_0^a\\&=iv\sqrt a ((\frac{2}{iii}.a^{\frac{iii}{2}}) - 0)\\&=\frac{8a^ii}{3}\end{align}\)
Therefore the surface area nether the curve enclosed past the parabola is \(\frac{8a^2}{three}\) square units.
Area Nether a Curve - Ellipse
The equation of the ellipse with the major axis of 2a and a minor axis of 2b is 102/atwo + y2/b2 = 1. This equation can exist transformed in the course as y = b/a .√(a2 - x2). Hither we calculate the surface 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 purlieus limits taken on the x-axis is from 0 to a. The calculations for the area of the ellipse are as follows.
\(\brainstorm{align}A &=iv\int_0^a y.dx \\&=4\int_0^4 \frac{b}{a}.{a^2 - x^ii}.dx\\&=\frac{4b}{a}[\frac{x}{ii}.\sqrt{a^2 - x^2} + \frac{a^ii}{2}Sin^{-ane}\frac{10}{a}]_0^a\\&=\frac{4b}{a}[(\frac{a}{2} \times 0) + \frac{a^2}{2}.Sin^{-1}1) - 0]\\&=\frac{4b}{a}.\frac{a^2}{two}.\frac{\pi}{2}\\&=\pi ab\end{align}\)
Therefore the area of the ellipse is πab sq units.
Area Nether The Curve - Between a Curve and A-Line
The area between a bend 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 aforementioned. The below figure shows the bend \(y_1\) = 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 utilise the boundaries to find the resultant area.
A = \(\int^b_a [f(10) - yard(x)].dx\)
Area Under a Curve - Between 2 Curves
The surface area between two curves can exist conveniently calculated past taking the difference of the areas of one curve from the area of another curve. Here the boundary with respect to the centrality 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 discover the area between these two curves. Here nosotros have the integral of the difference of the ii curves and utilize the boundaries to find the resultant.
A = \(\int^b_a [f(10) - g(10)].dx\)
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FAQs on Area Under The Bend
How to Find the Area Under the Curve?
The area under the bend can exist found using the process of integration or antiderivative. For this, we need the equation of the curve(y = f(ten)), the axis bounding the curve, and the purlieus limits of the curve. With this the area bounded under the bend tin can be calculated with the formula A = \(_a\int^b y.dx\)
What Are the Different Methods to Find the Expanse Under the Curve?
There are three wide methods to find the expanse under the curve. The area under the bend is calculated past dividing the area space into numerous pocket-sized rectangles, and so the areas are added to obtain the total expanse. The second method is to divide the area into a few rectangles and and then the areas are added to obtain the required expanse. The third method is to notice the area with the assistance of integration.
What Does Area Nether the Bend Mean?
The area under the bend means the surface area divisional by the bend, the axis, and the boundary points. The surface area under the curve is a two-dimensional area, which has been calculated with the assistance of the coordinate axes and by using the integration formula.
What Does Expanse Nether the Curve Stand for?
The area nether the curve represents the area enclosed under the curve and the centrality, which is marked with limiting points. This expanse under the bend gives the area of the irregular aeroplane shape in a two-dimensional array.
What Is Surface area Nether the Curve in Velocity Time Graph?
In the velocity-time graph, the velocity is graphed with respect to the y-centrality, and the fourth dimension is taken on the x-axis. With this, the area is assumed to be the product of velocity and fourth dimension and information technology gives the distance covered. Hence the surface area under the curve of the velocity-fourth dimension graph gives the altitude covered.
How to Interpret Surface area Under the Curve?
The area under the bend is the surface area between the curve and the coordinate centrality. Further boundaries are applied across the curve with respect to the axis to obtain the required surface area. The area under the curve is generally the area of irregular shapes that do not take any expanse formulas in geometry.
How to Summate Area Under the Curve Without Integration?
The area nether the curve can be calculated fifty-fifty without the use of integration. The area under the bend can be broken into smaller rectangles and so the summation of these areas gives the areas under the curve. Also another method is to pause the area under the curve into few rectangles, and then we can take the corresponding areas to obtain the area under the curve.
How to Judge the Area Under the Curve?
The area under the bend can be approximately calculated by breaking the area into small parts as modest rectangles. And the areas of these rectangles can be calculated and the summation of it gives the area under the curve. Another fashion to find the approximate area nether the bend is to draw a fix of few big rectangles so take a summation of their areas. Farther, we can merely find the exact surface area nether the curve with the assistance of definite integrals.
When to Use Area Under the Curve?
The area under the curve is useful to find the expanse of irregular shapes in a plane area. Nosotros mostly find formulas to discover the area of a circle, square, rectangle, quadrilaterals, polygon, simply we do not take any means to observe the surface area of irregular shapes. Here nosotros use the concept of definite integrals to obtain the surface area values.
When Is the Area Under the Curve Negative?
The area under the curve is negative if the curve is nether the axis or is in the negative quadrants of the coordinate axis. For this also the expanse of the curve is calculated using the normal method and a modulus is practical to the concluding answer. Even with the negative reply, only the value of the expanse is taken, without considering the negative sign of the answer.
Area Under The Curve Units,
Source: https://www.cuemath.com/calculus/area-under-the-curve/
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