Area Between Two Curves Calculator

what is Area between Curves?

In mathematics, the place among curves may be calculated with the difference between the definite vital of two factors or expressions. In -dimensional geometry, the place can explicit with the region covers by using the 2 one-of-a-kind curves. There are two capabilities required to calculate the region, f(x) and g(x) and the crucial limits from a to b wherein b ought to be greater than $a, b>a$ of the expression. $Y = f (x) \text{ where x} = a, x = b$ $Y = f (x)$ between limits of a and b

Method for location between Curves:

we can locate the areas among curves with the aid of using its widespread method if we have distinct curves $$m = f (x) & m = g (x)$$ Where $f (x) \text{ greater than} g (x)$ So the area bounded by two lines$x = a \text{ and} x = b$ is $$A = ∫ab[f (x) – g (x)] dx$$ So, the region between curves calculator computes the location wherein two curves intersect every different by means of the use of this fashionable method. however, an on-line integral Calculator lets in you to evaluate the integrals of the capabilities with appreciate to the variable worried.

Example:

Find the area between two curves $x^2 + 3y - x = 0$ and the straight line $x = 2y$.

Solution: The given equations are: 1. $x^2 + 3y - x = 0$ 2. $x = 2y$ **Step 1: Rewrite the first equation:** From $x^2 + 3y - x = 0$, solve for $y$: $$x^2 + 3y - x = 0$$ Add $x$ and subtract $x^2$ from both sides: $$x^2 + 3y - x + x - x^2 = x - x^2$$ $$3y = x - x^2$$ Divide by 3: $$y = \frac{x - x^2}{3}$$ **Step 2: Find the points of intersection:** Substitute $x = 2y$ into the first equation: $$x^2 + 3 \cdot \frac{x}{2} - x = 0$$ Simplify: $$x^2 + \frac{3x}{2} - x = 0$$ $$x^2 - \frac{x}{2} = 0$$ Factorize: $$x(x - \frac{1}{2}) = 0$$ Thus, $x = 0$ or $x = \frac{1}{2}$. **Step 3: Correlate $x$ values to find $y$:** Using $x = 2y$: - For $x = 0$, $y = 0$. - For $x = \frac{1}{2}$, $y = \frac{1}{4}$. The points of intersection are $P(0, 0)$ and $Q(\frac{1}{2}, \frac{1}{4})$. **Step 4: Set up the integral for the area:** The curve on the top is $f(x) = \frac{x}{2}$, and the curve on the bottom is $g(x) = \frac{x - x^2}{3}$. The area is: $$A = \int_{0}^{1/2} \left[ f(x) - g(x) \right] dx$$ Substitute the functions: $$A = \int_{0}^{1/2} \left[ \frac{x}{2} - \frac{x - x^2}{3} \right] dx$$ Simplify: $$A = \int_{0}^{1/2} \left[ \frac{3x}{6} - \frac{2x - 2x^2}{6} \right] dx$$ $$A = \int_{0}^{1/2} \left[ \frac{x + 2x^2}{6} \right] dx$$ **Step 5: Solve the integral:** $$A = \frac{1}{6} \int_{0}^{1/2} \left[ x + 2x^2 \right] dx$$ $$A = \frac{1}{6} \left[ \frac{x^2}{2} + \frac{2x^3}{3} \right]_{0}^{1/2}$$ Evaluate at $x = 1/2$: $$A = \frac{1}{6} \left[ \frac{(1/2)^2}{2} + \frac{2(1/2)^3}{3} \right]$$ $$A = \frac{1}{6} \left[ \frac{1}{8} + \frac{1}{12} \right]$$ Find the common denominator: $$A = \frac{1}{6} \cdot \frac{3 + 2}{24}$$ $$A = \frac{1}{6} \cdot \frac{5}{24}$$ $$A = \frac{5}{144}$$ The area is: $$A = \frac{5}{144}$$

FAQs

What is the Area Between Two Curves Calculator.

The Interval Boundary Calculator is a device assisting in determining the space covered by two graphs. To find this spot, we spot the difference between two line graphs' equations during a certain time period. - "Result" kept as is because it is a straightforward term meaning what is achieved. - "Gives" changed to "supplies" which means presenting or The apparatus simplifies determining this region with basic calculations, skipping the task of intricate calculations.

How does the Area Between Two Curves Calculator work.

Use the Area Between Two Curves Calculator by entering the formula of the curves and the specified domain. . s, we will use a calculator. Then, it will find the space between two lines by adding and subtracting the math shapes. The region's magnitude is usually determined through defined integrals, deducting the higher curve from the lower curve within the specified boundaries. The result is the area between the curves in the specified range.

How do I determine which surface is the greater line and which is the smaller line.

Typically, the curve above has higher y-values, while the curve below has smaller y-values. To tell apart two things, you can plot their shapes on a graph and look where each shape is placed above the other. Write 'If the curves intersect within the specified range, decomposition into separate segments may be required where each constitutes the upper segment. ' The calculator generally manages this process automatically when equipped with the fit operations and range.

What happens if the curves intersect within the interval.

If the graphs intersect within the specified range, the Area Between Two Curves Calculator divides the calculation into parts where one functions as the upper boundary and the other as the lower boundary. The whole thing will be figured out in bits where the upper line and lower line trade places. This guarantees that the geometric calculation includes the entire space enclosed between the curves, regardless of whether they intersect or cross each other over the defined range.

What is the significance of the area between two curves.

The gap between two lines is a basic idea in calculus, commonly applied in physics, engineering, economics, and various other disciplines. 'Various amounts may mean divergence in gathered measurements between dual factors, labor exercised over span by force, or comprehensive shifts in locomotion. ' In geometry, this method is used to calculate the space enclosed by two curves, which proves essential in scenarios dealing with optimization challenges and variations over time.

Can the calculator work with both functions and non-functions.

The space inside two shaped lines calculator works with math functions such as f(x) and h(x). If your trajectories are not explicitly formated (i. e. , they are not in the form of y = f(x)), the device may not be directly usable. However, when you deal with curves that are not directly written out, you may have to change the equation or use a calculator to find an estimate. The tool is best suitable for well-defined functions or equations.

Can I measure the space between curves in different dimensions, like for 3D shapes.

The Zone Between Two Plotted Lines Tool is typically designed for two-dimensional graphs that can be defined as functions of x and y. ** When dealing with three-dimensional shapes or surfaces, you would use certain formulas involving many variables and higher math. “This converter is not ideal for 3D conundrums, yet there are other instruments and devices adept at computing surface expansion and volumetric metrics in spatial models.

Can the location among two Curves be negative or not?

If both the curves lie on the x-axis, so the areas among curves can be bad (-). however, the signed price is the very last answer.

Why we use simplest specific necessary for finding the area Bounded with the aid of Curves?

The location between the curves calculator finds the location by way of distinctive functions simplest indefinite integrals due to the fact indefinite simply suggests the family of various features in addition to use to locate the region between two curves that integrate the difference of the expressions.