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 Calculator

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.

Area Between Two Curves Introduction
Area Between Two Curves Introduction

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).

Area Between Two Curves Example
Area Between Two Curves Example

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].

Area Between Two Curves Formula
Area Between Two Curves Formula

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.

Area Between Two Curves With Respect to Y Diagram
Area Between Two Curves With Respect to Y Diagram

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

Area Between Two Curves With Respect to Y
Area Between Two Curves With Respect to Y

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.

Area Between Two Compound Curves Diagram
Area Between Two Compound Curves Diagram

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 Compound Curves
Area Between Two Compound Curves

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.

Area Between Two Polar Curves Diagram
Area Between Two Polar Curves Diagram

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 Polar Curves
Area Between Two Polar Curves

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!

Area Between Two Curves Example 1
Area Between Two Curves Example 1
Area Between Two Curves Example 2
Area Between Two Curves Example 2

Frequently Asked Questions About Area Between Two Curves

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.