# Area Between Two Curves Calculator

The area between two curves calculator gives the area occupied between two curves using a free online tool. It is easy to use, and it displays its results in seconds thanks to STUDYQUERIES’s online area between two curves calculator.

## How to Use the Area Between Two Curves Calculator?

To use the area between the two curves calculator, follow these steps:

• Step 1: Enter the smaller function, the larger function, and the limit values in the given input fields
• Step 2: To calculate the area, click the Calculate Area button
• Step 3: Finally, in the new window, you will see the area between these two curves

## Area Between Two Curves

It is essential to integrate two curves in order to find the area between them. Using integration, we have learned to find the area under a curve. In the same way, we can also find the area between two intersecting curves using integration. In a given space, it would be the region that lies between two linear or non-linear curves.

It is also possible to find easily the composite area between two curves by modifying the formulas used for finding the area under two curves using integration. Please read on for a further discussion on this issue.

### Area Between Two Curves Introduction

Integral calculus can be used to calculate the area between two curves that fall between two intersecting curves. When we know the equation of two curves and the points at which they intersect, we can use integration to find the area under two curves.

According to the image, we have two functions, f(x) and g(x), and we need to find the area between these two curves. Using integration, we can easily calculate the area of the shaded area. We will discuss how to calculate this area in the next section.

### Area Between Two Curves Formula

We need to divide the area into many thin rectangular strips parallel to the y-axis starting from x = a to x = b in order to find the approximate area between two curves by integrating the areas of these thin strips. The rectangular strips will have a width of “dx” and a height of f(x) – g(x).

As a result, we can calculate the area of the triangle by using integration within the limits of a and b. The area between the two curves is dx(f(x) – g(x)). The formula for f(x) and g(x) is continuous on [a, b] and we have g(x) < f(x) for all x in [a, b].

General Conventions on Area Between Two Curves

• The upper function in a graph, i.e. the one with the larger y value for a given x, is called f(x).
• The lower function in the graph, that is, the one with a smaller value of y for a given x is called g(x).
• It is possible for two different regions on the graph to have different upper and lower functions. In such cases, it is important to calculate the area separately.
• Positive signs are assigned to the area above the y-axis.
• A negative sign is assigned to the area below the y-axis.

### Area Between Two Curves With Respect to Y

A curve whose equation is expressed in terms of y can be calculated by calculating the area between two curves with respect to the y-axis. Calculating the area along the y-axis is easier than calculating the area along the x-axis.

By applying integration, we add the areas of the horizontal strips to determine the area of the section between two curves by dividing the given rectangle into horizontal strips between the limits. Suppose f(y) and g(y) are continuous on [c, d], and g(y) < f(y) for all y in [c, d], then

General Conventions on Area Between Two Curves With Respect to Y

• Assume f(y) is the right function on the graph, that is, that which has the largest x value for the given y
• In the graph, the left function is the one with a smaller value of x for a given y, known as g(y).
• If there are different regions on the graph, different functions will be applied. If there are different regions on the graph, separate areas should be calculated for each one.
• Right on the x-axis, the area is assigned a positive sign.
• The negative sign is affixed to the area to the left of the x-axis.

### Area Between Two Compound Curves

The above-stated formulas will result in the incorrect calculation of areas between two compound curves that intersect with each other.

We calculated individual areas between curves between the different portions of the curves in each section of the curves shown in the image. If f(x) and g(x) are continuous in [a,b] intervals, then the area between the curves will be:

### Area Between Two Polar Curves

Also, where two polar curves meet, we can calculate their area through integral calculus. When we have two curves whose coordinates are not given in rectangular coordinates, but instead in polar coordinates, we use this method. To solve this problem, we can always convert the polar coordinates into rectangular coordinates, however, this method reduces the complexity of the problem.

Let us say we have two polar curves r0 = f(θ) and ri = g(θ)as shown in the image, and we want to find the area enclosed between these two curves such that α ≤ θ ≤ β where [α, β] is the bounded region. Therefore, the area between the curves will be:

### Area Between Two Curves Problems And Solutions

Now let’s solve some questions to learn how to use the general formula with the necessary conventions!

## Find Area Between Two Curves Calculator:

The “Find Area Between Two Curves Calculator” is a tool that helps determine the area enclosed between two curves. It is commonly used in calculus to find the region bounded by two curves in a given interval. The calculator requires the user to input the equations of the two curves and specify the interval of interest. It then calculates the area between the curves by integrating the difference between the two functions over the specified interval.

For example, let’s consider the curves y = x^2 and y = 2x – 1. To find the area between these two curves, you would input these equations into the calculator and specify the interval over which you want to find the area. The calculator would then calculate the points of intersection of the two curves, typically by setting the equations equal to each other and solving for x. In this case, the curves intersect at x = -1 and x = 2. Finally, the calculator integrates the difference between the two functions, y = x^2 – (2x – 1), over the interval [-1, 2] to find the area between the curves.

## Area Between Two Curves Calculator With Steps:

The “Area Between Two Curves Calculator With Steps” is a more comprehensive version of the previous calculator. In addition to providing the numerical result, this calculator also shows the step-by-step process of finding the area between two curves. It helps users understand the calculations involved and provides a detailed explanation of each step.

Using the example of the curves y = x^2 and y = 2x – 1, the calculator with steps would start by determining the points of intersection by setting the two equations equal to each other: x^2 = 2x – 1. Solving this equation gives x = -1 and x = 2. Then, the calculator would integrate the difference between the two functions, y = x^2 – (2x – 1), over the interval [-1, 2] step by step. It would break down the integration process and explain each step, such as finding the antiderivative and evaluating the integral limits. This detailed approach allows users to follow along and understand the underlying concepts.

## Area Between Two Polar Curves Calculator:

The “Area Between Two Polar Curves Calculator” is designed specifically for calculating the area enclosed between two polar curves. In polar coordinates, curves are represented by equations involving angles (θ) and radii (r). This calculator takes the equations of the two polar curves and determines the area enclosed between them.

For example, let’s consider the polar curves r = 2sin(θ) and r = 4cos(θ). To find the area between these two curves, you would input their respective equations into the calculator. The calculator would then find the points of intersection of the two curves by equating the radii: 2sin(θ) = 4cos(θ). Solving this equation gives the points of intersection. Finally, the calculator computes the area between the curves using polar integration techniques. It integrates the difference between the two equations, r = 2sin(θ) – 4cos(θ), with respect to θ over the interval where the curves intersect.

## Area Of Region Between Two Curves Calculator:

The “Area Of Region Between Two Curves Calculator” is similar to the “Find Area Between Two Curves Calculator” but with a slight difference. Instead of specifying the interval explicitly, this calculator automatically detects the interval by finding the points of intersection of the curves. It calculates the area of the entire region enclosed between the curves without requiring the user to manually determine the interval.

For instance, consider the curves y = x^2 and y = 2x – 1 again. The calculator would automatically find the points of intersection by equating the two equations: x^2 = 2x – 1. Solving this equation gives x = -1 and x = 2. Then, the calculator calculates the area between the curves by integrating the difference between the two functions, y = x^2 – (2x – 1), over the interval [-1, 2], which is automatically determined.

## Area Between Two Curves Formula:

The “Area Between Two Curves Formula” is not a calculator itself but rather a tool that provides the formula for calculating the area between two curves. The formula depends on whether the curves are given as functions of x or y.

For curves defined by y = f(x) and y = g(x), the formula is:
Area = ∫[a, b] (f(x) – g(x)) dx.

If the curves are given as functions of y, the formula is:
Area = ∫[c, d] (g(y) – f(y)) dy.

These formulas represent the signed area between the curves, meaning that the area can be negative if one curve is above the other in certain regions.

## Area Between Two Curves Calculator With Respect To Y:

The “Area Between Two Curves Calculator With Respect To Y” is similar to the first calculator, but it integrates with respect to y instead of x. This calculator is useful when the curves are better expressed as functions of y.

For example, consider the curves x = y^2 and x = y. To find the area between these curves with respect to y, you would input their respective equations into the calculator. The calculator would find the points of intersection by equating the x-values: y^2 = y. Solving this equation gives y = 0 and y = 1. Finally, the calculator integrates the difference between the two functions, x = y^2 – y, with respect to y over the interval [0, 1] to find the area between the curves.

## Volume Of Area Between Two Curves Calculator:

The “Volume Of Area Between Two Curves Calculator” goes beyond calculating the area between two curves and focuses on finding the volume of the solid formed by rotating the enclosed region about an axis. It uses the disk or washer method to determine the volume.

The calculator requires the user to input the equations of the curves, the interval, and the axis of rotation. It then calculates the volume by integrating the cross-sectional areas of the slices obtained by rotating the region between the curves. The disk or washer method depends on whether the slices are discs or washers, depending on the shape of the region.

For example, let’s consider rotating the region between y = x^2 and y = 2x – 1 about the x-axis. The calculator would find the volume of the solid formed by rotating this region. It would calculate the cross-sectional areas of the discs or washers obtained by rotating the region and then integrate those areas over the specified interval to find the volume.

## Area Between 4 Curves Calculator:

The “Area Between 4 Curves Calculator” extends the concept of finding the area between two curves to finding the area between four curves. It allows users to compute the area enclosed by four curves in a given interval.

To use this calculator, the user inputs the equations of the four curves and specifies the interval of interest. The calculator then calculates the area between the four curves by integrating the difference between the outermost and innermost functions over the specified interval.

## Area Under The Curve Calculator:

The “Area Under The Curve Calculator” is a tool specifically designed to calculate the area between a curve and the x-axis. It is commonly used in

calculus to determine the definite integral of a function over a given interval.

To use this calculator, the user inputs the equation of the curve and specifies the interval of interest. The calculator then calculates the area under the curve by integrating the function over the specified interval.

For example, consider the curve y = x^2. To find the area under this curve between x = 0 and x = 2, you would input the equation into the calculator and specify the interval. The calculator would then integrate the function, y = x^2, with respect to x over the interval [0, 2] to find the area under the curve.

To find the area between two curves using a calculator, there are several approaches you can take depending on the type of calculator you have. Here are explanations for each of the scenarios you mentioned:

What does the area between two curves represent?

You will then be able to determine the displacement between the initial and final positions of the particles by calculating the areas between the two graphs.

Is the area between two curves always positive?

And finally, the area between two curves always has a positive value, unlike the area under a curve we studied in the previous chapter.

What is the first step toward finding the area between two curves?

You will first take the integrals of both curves. Once you have taken the integrals of both curves, solve them normally. Finally, you will subtract the integral of the curve higher on the graph from the integral of the curve lower on the graph.

How do you use integration to find areas?

By doing a definite integral between two points, you can find the area under a curve between two points. Using a limit between a and b, integrate y = f(x) between a and b to find the area under the curve y = f(x). Negative values will be displayed for areas below the x-axis and positive values for areas above the x-axis.

Why are two curves positive?

When solving that type of problem, one approaches each piece individually, integrating the (top function) and (bottom function) to guarantee a positive (nonnegative) result for each piece. “Area between two graphs” is, by definition, positive regardless of where it lies in the plane.

How do you find the area between two curves in Excel?

• Type “LINEST(yrange, range, TRUE, FALSE)” into a blank cell in your Excel spreadsheet.
• Change the options in the LINEST command to reflect the location of your data for one of the graphs.
• Press “Enter” to get the equation for the first set of data.

Can the area between two curves be negative?

It is possible for a definite integral to be negative. When all the area within the interval is below the x-axis but still above the curve, then the result is negative.

How do you find the area of a vertical slice?

Area Between Two Curves Using Vertical Slices A = ∫ a b ( g ( x ) − f ( x ) ) dx.

How To Find The Area Between Two Curves Calculator?

To find the area between two curves using a calculator, you would typically follow these steps:

Step 1: Enter the equations of the two curves into your calculator. Make sure the equations are in a form that your calculator can handle. For example, if you have a graphing calculator, you may need to solve the equations for y explicitly.

Step 2: Determine the points of intersection between the two curves. Set the equations equal to each other and solve for the x-values or y-values where they intersect. This can be done by finding the zeros or using an appropriate intersection function on your calculator.

Step 3: Determine the interval over which you want to find the area. This interval is typically defined by the points of intersection found in Step 2.

Step 4: Use the integral function on your calculator to evaluate the definite integral of the absolute difference between the two curves over the specified interval. The integral function may vary depending on the type of calculator you have.

Step 5: Once you’ve entered the integral, evaluate it to find the area between the two curves. The result will be displayed on your calculator.

How To Find Area Between Two Curves On Graphing Calculator Ti84?

If you have a TI-84 graphing calculator, you can follow these steps to find the area between two curves:

Step 1: Enter the equations of the two curves into your calculator. You can do this by selecting the “Y=” function and entering the equations as functions of x.

Step 2: Graph the two equations by pressing the “GRAPH” button.

Step 3: Determine the points of intersection between the two curves. To do this, use the “2nd” button followed by the “TRACE” button and select the “Intersection” option. This will allow you to find the x-values or y-values where the curves intersect.

Step 4: Determine the interval over which you want to find the area. This interval is typically defined by the points of intersection found in Step 3.

Step 5: Use the “2nd” button followed by the “CALC” button to access the “Math” menu. Select the “7: fnInt” option to calculate the definite integral.

Step 6: Enter the absolute difference between the two curves as the function to integrate. Specify the variable, the lower and upper limits of integration (defined by the interval), and press “ENTER”.

Step 7: The calculator will evaluate the integral and display the result, which represents the area between the two curves.

How To Calculate Area Between Two Curves On Calculator?

If you have a scientific calculator that does not have graphing capabilities, you can use the following steps to calculate the area between two curves:

Step 1: Determine the points of intersection between the two curves by setting the equations equal to each other and solving for the x-values or y-values where they intersect.

Step 2: Determine the interval over which you want to find the area. This interval is typically defined by the points of intersection found in Step 1.

Step 3: Use the integral function on your calculator to evaluate the definite integral of the absolute difference between the two curves over the specified interval. Enter the integrand as a function and specify the variable, the lower and upper limits of integration (defined by the interval).

Step 4: Evaluate the integral to find the area between the two curves. The result will be displayed on your calculator.

How To Find Area Between Two Curves Using Calculator?

To find the area between two curves using a basic calculator without graphing capabilities, you can follow these steps:

Step 1: Determine the points of intersection between the two curves by setting the equations equal to each other and solving for the x-values or y-values where they intersect.

Step 2: Determine the interval over which you want to find the area. This interval is typically defined by the points of intersection found in Step 1.

Step 3: Divide the interval into small subintervals and approximate the area between the curves by treating each subinterval as a rectangle. Calculate the width of each rectangle as the difference between the x-values or y-values at the boundaries of the subinterval. Calculate the height of each rectangle as the difference between the y-values or x-values of the corresponding curves at the midpoint of the subinterval.

Step 4: Sum up the areas of all the rectangles to approximate the total area between the curves.

Keep in mind that using a graphing calculator or specialized software can provide more accurate and efficient calculations for finding the area between two curves.