Area Under The Curve -Types, Formulas, Solved Examples, and FAQs

Using definite integration to find area of a circle with radius r
Using definite integration to find area of a circle with radius r

Area Under the Curve is the area enclosed by the curve and the coordinate axes, it is calculated by taking very small rectangles and then taking their sum if we take infinitely small rectangles then their sum is calculated by taking the limit of the function so formed. We use the area under the curve to find the area of the figure that is not symmetric and can not be calculated using regular means.

In this article, we will learn about, the area under the curve, its applications, examples, and others in detail.

What is the Area Under the Curve?

The Area Under the Curve is the area enclosed by any curve with the x-axis and given boundary conditions i.e., the area bounded by function y = f(x), x-axis, and the line x = a, and x = b. In some cases, there is only one or no boundary condition as the curve intersects the x-axis either once or twice respectively.

The area under the curve can be calculated using various methods such as Reimann sum, and Definite integral and also we can also approximate the area using the basic shapes i.e., triangle, rectangle, trapezium, etc.

Calculating the Area Under the Curve

To calculate the area under the curve, we can use the following methods such as:

  • Using Riemann Sums
  • Using Definite Integrals
  • Using Approximation

Let’s study these methods in detail as follows:

Using Riemann Sums

The Reimann sum is calculated by dividing a given function’s graph into smaller rectangles and summing the areas of each rectangle. The more rectangles we consider by subdividing the provided interval, the more precise the area computed by this approach is; nevertheless, the more subintervals we consider, the more difficult the calculations get.

Reimann Sum can be classified into three more categories such as:

  • Left Reimann Sum
  • Right Reimann Sum
  • Midpoint Reimann Sum

The area using the Reimann sum is given as follows:

\bold{Area = \sum_{i=1}^{n}f(x_i)\Delta x_i}

Where,

  • f(xi) is the value of the function being integrated at the ith sample point
  • Δx = (b-a)/n is the width of each subinterval,
    • a and b are the limits of integration and
    • n is the number of subintervals
  • ∑ represents the sum of all the terms from i=1 to n,

Example: Find the area under the curve for function, f(x) = x2 between the limits x = 0 and x = 2.

Solution:

We want to find the area under the curve of this function between x = 0 and x = 2. We will use a left Riemann sum with n = 4 subintervals to approximate the area.

Let’s calculate the area under the curve using 4 subintervals.

Thus, width of subintervals, Δx = (2-0)/4 = 0.5

All the 4 subintervals are,

a = 0 = x0 < x1 <x2 < x3 < x4 = 2 = b

x0 = 0, x1 = 0.5, x2 = 1, x3 = 1.5, x4 = 2

Now we can evaluate the function at these x-values to find the heights of each rectangle:

f(x0) = (0)2 = 0
f(x1) = (0.5)2 = 0.25
f(x2) = (1)2 = 1
f(x3) = (1.5)2 = 2.25
f(x4) = (2)2 = 4

The area under the curve can now be approximated by summing the areas of the rectangles formed by these heights:

A ≈ Δx[f(x0) + f(x1) + f(x2) + f(x3)] = 0.5[0 + 0.25 + 1 + 2.25] = 1.25

Therefore, the area under the curve of f(x) = x2 between x = 0 and x = 2, approximated using a left Riemann sum with 4 subintervals, is approximately 1.25.

Using Definite Integrals

Definite Integral is the almost same as the Reimann sum but here the number of subintervals approaches infinity. If the function is given for interval [a, b] then definite integral is defined as:

 \int_{a}^{b} f(x) dx = \lim_{n\to \infty}\sum_{i=1}^{n}f(x_i)\Delta x_i

A definite Integral gives the exact area under the curve, unlike the Reimann sum. The definite integral is calculated by finding the antiderivative of the function and evaluating it at the limits of integration.

Area with Respect to X-Axis

The curve shown in the image below is represented using y = f(x). We need to calculate the area under the curve with respect to the x-axis. The boundary values for the curve on the x-axis are a and b respectively. The area A under this curve with respect to the x-axis is calculated between the points x = a and x = b. Consider the following curve:

The formula for Area Under the Curve w.r.t to the x-axis is given by:

\bold{A = \int_{a}^{b}y.dx}

or

\bold{A = \int_{a}^{b}f(x)dx}

Where,

  • A is the area under the curve,
  • y or f(x) is the equation of the curve
  • a, and b are the x-values or limit of integration, for which we need to calculate the area.

Area with Respect to Y-Axis

The curve shown in the image above is represented using x = f(y). We need to calculate the area under the curve with respect to Y-axis. The boundary values for the curve on the Y-axis are a and b respectively. The area A under this curve with respect to Y-axis between the points y = a and y = b. Consider the following curve:

The formula for Area Under the Curve w.r.t to the y-axis is given by:

\bold{A = \int_{a}^{b}x.dy}

or

\bold{A = \int_{a}^{b}f(y)dy}

Where,

  • A is the area under the curve,
  • x or f(y) is the equation of the curve, and
  • a, b are the y-intercepts.

Learn more, Area Between Two Curves

Approximating the Area Under The Curve

Approximating the area under the curve involves using simple geometric shapes, such as rectangles or trapezoids, to estimate the area under the curve. This method is useful when the function is difficult to integrate or when it is not possible to find an antiderivative of the function. The accuracy of the approximation depends on the size and number of the shapes used.

Calculating Area Under the Curve

We can easily calculate the area of the various curve using the concepts discussed in the given article. Now let’s consider some examples of calculation of Area Under the Curve for some common curves.

Area Under the Curve: Parabola

We know that a standard parabola is divided into two symmetric parts by either the x-axis or the y-axis. Suppose we take a parabola y2 = 4ax and then its area is to be calculated from x = 0 to x = a. And if required we double its area to find the area of the parabola in both the quadrant.

Calculating area,

y2 = 4ax

y = √(4ax)

A = 2∫0a y.dx

A = 2∫0a√(4ax).dx

A = 4√(a)∫0a√(x).dx

A = 4√(a){2/3.a3/2}

A = 8/3a2

Thus, the area under the parabola from x = 0 to x = a is 8/3a2 square units

Area Under the Curve: Circle

A circle is a closed curve whose circumference is always at an equal distance from its center. Its area is calculated by first calculating the area in the first quadrant and then multiplying it by 4 for all four quadrants.

Suppose we take a circle x2 + y2 = a2 and then its area is to be calculated from x = 0 to x = a in the first quadrant. And if required we quadruple its area to find the area of the circle.

Calculating area,

x2 + y2 = a2

y = √(a2 – x2).dx

A = 4∫0a y.dx

A = 4∫0a√(a2 – x2).dx

A = 4[x/2√(a2 – x2) + a2/2 sin-1(x/a)]a0

A = 4[{(a/2).0 + a2/2.sin-1} – 0]

A = 4(a2/2)(π/2)

A = πa2

Thus, the area under the circle is πa2 square units

Area Under the Curve: Ellipse

A circle is a closed curve. Its area is calculated by first calculating the area in the first quadrant and then multiplying it by 4 for all four quadrants.

Suppose we take a circle (x/a)2 + (y/b)2 = 1 and then its area is to be calculated from x = 0 to x = a in the first quadrant. And if required we quadruple its area to find the area of the ellipse.

Calculating area,

(x/a)2 + (y/b)2 = 1

y = b/a√(a2 – x2).dx

A = 4∫0a y.dx

A = 4b/a∫0a√(a2 – x2).dx

A = 4b/a[x/2√(a2 – x2) + a2/2 sin-1(x/a)]a0

A = 4b/a[{(a/2).0 + a2/2.sin-1} – 0]

A = 4b/a(a2/2)(π/2)

A = πab

Thus, the area under the ellipse is πab square units.

Area Under the Curve Formula

The formula for various types of calculation of Area Under the Curve is tabulated below:

Type of Area

Formula of Area

Area Using Riemanns Sum \bold{Area = \sum_{i=1}^{n}f(x_i)\Delta x_i}
Area with Respect to y-axis \bold{A = \int_{a}^{b}f(y)dy}
Area with respect to x-axis \bold{A = \int_{a}^{b}f(x)dx}
Area under parabola 2∫ab√(4ax).dx
Area under Circle 4∫ab√(a2 – x2).dx
Area under Ellipse 4b/a∫ab√(a2 – x2).dx

Also, Read

Sample Examples on Area Under Curve

Example 1: Find the area under the curve y2 = 12x and the X-axis.

Solution:

The given curve equation is y2 = 12x.

This is an equation of parabola with a = 3 so, y2 = 4(3)(x)

The graph for the required area is shown below:

The X-axis divides the above parabola into 2 equal parts. So, we can find the area in the first quadrant and then multiply it by 2 to get the required area.

So, we can find the required area as:

A = 2\int_{a}^{b}ydx

A = 2\int_{0}^{3}\sqrt{12x}dx

A = 2\sqrt{12}[\frac{2x^\frac{3}{2}}{3}]_0^3

A = \frac{4\sqrt{12}}{3}[x^\frac{3}{2}]_0^3

A = \frac{4\sqrt{12}}{3}*\sqrt{27}

⇒ A = 24 sq. units

Example 2: Calculate the area under the curve x = y3 – 9 between the points y = 3 and y = 4.

Solution:

Given, the equation of curve is x = y3 – 9.

The boundary points are (0, 3) and (0, 4) .

As the equation of curve is of the form x = f(y) and the points are also on the Y-axis, we will use the formula,

A = \int_{a}^{b}x.dy

A = \int_{3}^{4}(y^3-9)dy

A = [\frac{y^4}{4}-9y]^4_3

A = (64-36)-(\frac{81}{4}-27)

A = 28+\frac{27}{4}

⇒ A = 139/4 sq. units

Example 3: Calculate the area under the curve y = x2 – 7 between the points x = 5 and x = 10.

Solution:

Given, the curve is y = x2−7 and the boundary points are (5, 0) and (10, 0)

Thus, the area under the curve is given by:

A = \int_{5}^{10}(x^2-7)dx

A = [\frac{x^3}{3}-7x]_5^{10}

⇒ A = (100/3 – 70) – (125/3 – 35)

⇒ A = 790/3 – 23/3

⇒ A = 770/3 sq. units

Example 4: Find the area enclosed by the parabola y2 = 4ax and the line x = a in the first quadrant.

Solution:

The curve and the line given can be drawn as follows:

Now, the equation of curve is y2 = 4ax.

The boundary points come out to be (0,0) and (a,0).

So the area with respect to X-axis can be calculated as:

A=\int_{0}^{a}ydx

A=\int_{0}^{a}\sqrt{4ax}dx

A=[\sqrt{4a}\frac{x^{\frac{1}{2}+1}}{\frac{3}{2}}]_0^a

A=2×\frac{2}{3}\sqrt{a}[x^{\frac{3}{2}}]_0^a

A=\frac{4\sqrt{a}}{3}×a^\frac{3}{2}

A=\frac{4a^2}{3} sq. units

Example 5: Find the area covered by the circle x2 + y2 = 25 in the first quadrant.

Solution:

Given, x2 + y2 = 25.

The curve can be drawn as:

The required area has been shaded in the above figure. From the equation we can see that radius of the circle is 5 units.

As, x2 + y2 = 25

y = \sqrt{25-x^2}

To find the area, we shall use:

A = \int_{a}^{b}ydx

A = \int_{0}^{5}\sqrt{25-x^2}dx

A = [\frac{x}{2}(\sqrt{25-x^2}+\frac{25}{2}sin^{-1}\frac{x}{5})]_0^5

A = [(\frac{5}{2}×0 +\frac{25}{2}sin^{-1}(1))-0]

A = \frac{25}{2}×\frac{\pi}{2}

⇒ A = 25 π/4 sq. units

FAQs on Area Under The Curve

Q1: Define Area Under a Curve.

Answer:

The region enclosed by the curve, the axis, and the boundary points is referred to as the area under the curve. Using the coordinate axes and the integration formula, the area under the curve has been determined as a two-dimensional area.

Q2: How to Calculate Area Under a Curve?

Answer:

There are three methods to find the area under the curve, that are:

  • Riemann Sums involve dividing the curve into smaller rectangles and adding their areas, with the number of subintervals affecting the precision of the result.
  • Definite Integrals are similar to Riemann sums but use an infinite number of subintervals to provide an exact result.
  • Approximation Methods is used known geometric shapes to approximate the area under the curve.

Q3: What is the Difference Between a Definite Integral and a Riemann sum?

Answer:

The key difference between a definite integral and a Riemann sum is that a definite integral represents the exact area under a given curve whereas a Riemann sum represents the approximate value of the area and the accuracy of the sum depends upon the chosen partition size.

Q4: Can Area Under Curve be Negative?

Answer:

If the curve is below the axis or lies in the coordinate axis’s negative quadrants, the area under the curve is negative. In this case as well, the area under the curve is computed using the conventional approach, and the solution is then modulated. Even in cases when the answer is negative, just the area’s value is taken into account, not the answer’s negative sign.

Q5: How do you Interpret the Sign of the Area Under a Curve?

Answer:

The sign of the area shows that the area under the curve is above the x-axis or below the x-axis. If the area is positive, then area under the curve is above the x-axis and if it negative then area under the curve is below the x-axis.

Q6: How is the Area Under the Curve Approximated?

Answer:

By segmenting the region into tiny rectangles, the area under the curve may be roughly estimated. And by adding the areas of these rectangles, one may get the area under the curve. A collection of a few large rectangles may be drawn, and their areas can then be added to determine the approximate area under the curve. Additionally, with the use of definite integrals, we can easily determine the precise area under the curve.

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