How To Find The Area Of A Parabola
Expanse Under the Curve
Area under the curve is calculated past dissimilar methods, of which the antiderivative method of finding the surface area is near popular. The area nether the curve can exist constitute past 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 equally square, rectangle, quadrilateral, polygon, circle, simply there is no divers formula to notice the area under the curve. The procedure of integration helps to solve the equation and find the required surface area.
For finding the areas of irregular plane surfaces the methods of antiderivatives are very helpful. Here nosotros shall larn how to find the area under the bend with respect to the axis, to notice the expanse between a bend and a line, and to find the area between two curves.
| 1. | How to Find Expanse Nether The Curve? |
| 2. | Different Methods to Find Expanse Under The Curve |
| 3. | Formula For Area Under The Bend |
| iv. | Area Under The Bend - Circle |
| 5. | Area Under The Curve - Parabola |
| 6. | Expanse Nether The Curve - Ellipse |
| 7. | Area Between a Bend and A-Line |
| 8. | Area Betwixt Ii Curves |
| 9. | Solved Examples |
| 10. | Practice Questions |
| xi. | FAQs on Area Nether The Curve |
How to Observe Area Nether The Curve?
The area nether the curve tin can be calculated through three uncomplicated steps. Kickoff, we need to know the equation of the curve(y = f(x)), the limits across which the expanse is to be calculated, and the axis enclosing the area. Secondly, we have to find the integration (antiderivative) of the curve. Finally, we demand to apply the upper limit and lower limit to the integral respond and take the difference to obtain the area under the curve.
Surface area = \(_a\int^b y.dx \)
= \(_a\int^b f(ten).dx\)
=\( [g(ten)]^b_a\)
=\( g(b) - g(a)\)
Different Methods to Find Area Under The Curve
The expanse under the curve can be computed using three methods. Also, the method used to find the area under the bend depends on the need and the available data inputs, to find the surface area under the curve. Here nosotros shall await into the below iii methods to discover the area under the curve.
Method - I: Here the surface area nether the curve is cleaved downwards 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 upward to infinity. The formula for the total surface area under the curve is A = \(\lim_{x \rightarrow \infty}\sum _{i = 1}^nf(x).\delta x\).
Method - 2: This method as well uses a similar procedure equally the above to find the area nether the curve. Hither the surface area under the curve is divided into a few rectangles. Further, the areas of these rectangles are added to get the surface area under the curve. This method is an easy method to notice the area nether the curve, only it merely provides an approximate value of the area under the curve.
Method - Iii: This method makes utilise of the integration procedure to detect the area under the curve. To find the area nether the bend past this method integration nosotros demand the equation of the bend, the noesis of the bounding lines or centrality, and the boundary limiting points. For a bend having an equation y = f(x), and bounded by the x-centrality and with limit values of a and b respectively, the formula for the surface area under the curve is A = \( _a\int^b f(x).dx\)
Formula For Area Under the Curve
The area of the bend can exist calculated with respect to the different axes, every bit the boundary for the given curve. The surface area under the curve can exist 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: Hither we shall first await at the area enclosed by the curve y = f(x) and the ten-axis. The below figures presents the expanse 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 10-centrality is A = \(_a\int^b f(x).dx\)
Area with respect to the y-axis: The expanse of the bend bounded past the curve x = f(y), the y-axis, beyond 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 bend 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-centrality and bounded past the x-axis is obtained by taking the limits a and b. The formula for the surface area to a higher place the curve and the 10-centrality is every bit follows.
A = |\(_a\int ^bf(x).dx\)|
Surface area in a higher place and beneath the axis: The area of the bend which is partly beneath the axis and partly higher up the axis is divided into ii 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 starting time calculating the area of the part of the circle in the outset quadrant. Hither the equation of the circumvolve xii + ytwo = a2 is changed to an equation of a curve as y = √(a2 - 102). This equation of the curve is used to find the expanse with respect to the x-centrality and the limits from 0 to a.
The area of the circumvolve is iv times the surface area of the quadrant of the circumvolve. The area of the quadrant is calculated by integrating the equation of the curve across the limits in the showtime quadrant.
A = 4\(\int^a_0 y.dx\)
= 4\(\int^a_0 \sqrt{a^2 - x^ii}.dx\)
= 4\([\frac{ten}{2}\sqrt{a^2 - x^two} + \frac{a^2}{2}Sin^{-one}\frac{x}{a}]^a_0\)
= 4[((a/2)× 0 + (a2/two)Sin-i1) - 0]
= four(atwo/two)(π/2)
= 2πr
Hence the area of the circle is πa2 square units.
Surface area Under a Curve - Parabola
A parabola has an axis that divides the parabola into two symmetric parts. Here nosotros take a parabola that is symmetric forth the x-axis and has an equation y2 = 4ax. This can be transformed every bit y = √(4ax). We showtime observe the expanse of the parabola in the showtime quadrant with respect to the x-axis and along the limits from 0 to a. Hither we integrate the equation inside the purlieus and double information technology, to obtain the expanse of the whole parabola. The derivations for the area of the parabola is as follows.
\(\begin{marshal}A &=ii \int_0^a\sqrt{4ax}.dx\\ &=iv\sqrt a \int_0^a\sqrt x.dx\\& =4\sqrt a[\frac{ii}{3}.x^{\frac{three}{ii}}]_0^a\\&=4\sqrt a ((\frac{2}{3}.a^{\frac{3}{2}}) - 0)\\&=\frac{8a^two}{3}\end{align}\)
Therefore the area under the curve enclosed by the parabola is \(\frac{8a^2}{3}\) square units.
Expanse Under a Curve - Ellipse
The equation of the ellipse with the major centrality of 2a and a minor axis of 2b is x2/a2 + y2/b2 = one. This equation can exist transformed in the grade every bit y = b/a .√(a2 - tentwo). Hither we summate the area bounded by the ellipse in the start coordinate and with the ten-axis, and farther multiply it with four to obtain the area of the ellipse. The boundary limits taken on the 10-axis is from 0 to a. The calculations for the area of the ellipse are as follows.
\(\begin{align}A &=4\int_0^a y.dx \\&=iv\int_0^iv \frac{b}{a}.{a^two - x^2}.dx\\&=\frac{4b}{a}[\frac{x}{2}.\sqrt{a^2 - x^ii} + \frac{a^2}{2}Sin^{-i}\frac{ten}{a}]_0^a\\&=\frac{4b}{a}[(\frac{a}{2} \times 0) + \frac{a^2}{2}.Sin^{-1}i) - 0]\\&=\frac{4b}{a}.\frac{a^2}{ii}.\frac{\pi}{2}\\&=\pi ab\end{align}\)
Therefore the area of the ellipse is πab sq units.
Area Under The Bend - Between a Curve and A-Line
The expanse betwixt a bend and a line can be conveniently calculated by taking the difference of the areas of 1 curve and the area under the line. Here the boundary with respect to the axis for both the bend and the line is the same. The below effigy shows the curve \(y_1\) = f(10), and the line \(y_2\) = g(x), and the objective is to observe the area between the curve and the line. Here we accept the integral of the difference of the 2 curves and apply the boundaries to find the resultant area.
A = \(\int^b_a [f(x) - g(x)].dx\)
Area Under a Bend - Between Ii Curves
The area between 2 curves can be conveniently calculated by taking the difference of the areas of one curve from the expanse of another bend. 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 \(y_2\) = k(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 utilise the boundaries to find the resultant.
A = \(\int^b_a [f(10) - m(x)].dx\)
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FAQs on Area Under The Bend
How to Observe the Area Under the Curve?
The area nether the curve can be found using the procedure of integration or antiderivative. For this, nosotros need the equation of the curve(y = f(x)), the axis bounding the curve, and the boundary limits of the bend. With this the area divisional under the bend can exist calculated with the formula A = \(_a\int^b y.dx\)
What Are the Different Methods to Find the Area Nether the Curve?
There are three broad methods to notice the area under the curve. The area under the curve is calculated by dividing the expanse infinite into numerous small rectangles, then the areas are added to obtain the total area. The second method is to carve up the area into a few rectangles then the areas are added to obtain the required surface area. The tertiary method is to find the expanse with the help of integration.
What Does Area Under the Curve Mean?
The expanse under the curve means the area bounded by the curve, the axis, and the purlieus points. The expanse under the bend is a two-dimensional area, which has been calculated with the help of the coordinate axes and past 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 Bend in Velocity Time Graph?
In the velocity-time graph, the velocity is graphed with respect to the y-centrality, and the time is taken on the x-axis. With this, the expanse is causeless to be the product of velocity and time and it gives the distance covered. Hence the area under the bend of the velocity-time graph gives the altitude covered.
How to Translate Expanse Under the Curve?
The area nether the curve is the area between the curve and the coordinate axis. Farther boundaries are applied across the curve with respect to the axis to obtain the required area. The area under the bend is mostly the area of irregular shapes that do not have any area formulas in geometry.
How to Calculate Area Under the Curve Without Integration?
The surface area nether the curve can be calculated even without the utilise of integration. The area under the curve can exist broken into smaller rectangles and and then the summation of these areas gives the areas nether the curve. Also another method is to break the expanse under the curve into few rectangles, so we tin can have the corresponding areas to obtain the surface area under the bend.
How to Approximate the Area Under the Bend?
The area under the curve can be approximately calculated by breaking the surface area into minor parts as small rectangles. And the areas of these rectangles can exist calculated and the summation of it gives the expanse under the bend. Another fashion to find the approximate area under the curve is to draw a set of few large rectangles so take a summation of their areas. Farther, nosotros can simply find the exact expanse under the curve with the aid of definite integrals.
When to Utilise Area Under the Curve?
The area under the curve is useful to find the area of irregular shapes in a plane surface area. Nosotros generally notice formulas to find the surface area of a circle, square, rectangle, quadrilaterals, polygon, but we do not have any means to detect the area of irregular shapes. Here we use the concept of definite integrals to obtain the expanse values.
When Is the Surface area Under the Curve Negative?
The area under the curve is negative if the bend is nether the centrality 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 reply, only the value of the surface area is taken, without considering the negative sign of the respond.
Source: https://www.cuemath.com/calculus/area-under-the-curve/
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