What is the Area Under the Curve?

The area under the curve refers to the region enclosed by the graph of a function, the x-axis, and the vertical lines at the boundaries of a given interval. It is a key concept in integral calculus used to quantify the accumulation of a quantity represented by the function.

What Methods Can Be Used to Find the Area Under the Curve?

There are several methods to find the area under the curve, including: Definite Integration: The exact area using the integral. Riemann Sums: Approximating the area by summing the areas of rectangles under the curve. Trapezoidal Rule: Approximating the area using trapezoids. Simpson's Rule: Using parabolic segments for more accurate approximation.

How Do You Find the Area Between a Curve and a Line?

To find the area between a curve and a line: Determine the points of intersection between the curve and the line. Set up the integral of the difference between the curve function and the line function over the interval of intersection. Evaluate the integral to find the area.

How Do You Calculate the Area Between Two Curves?

To calculate the area between two curves: Identify the points where the curves intersect. Set up the integral of the difference between the upper curve and the lower curve over the interval of intersection. Evaluate the integral to determine the area.

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Area Under the Curve

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Area Under the Curve: Definition and Importance

The area under the curve is a concept in integral calculus that quantifies the total area enclosed by a curve, the x-axis, and specified vertical boundaries on the graph of a function. This area is calculated using definite integrals and represents the accumulation of a quantity described by the function over a given interval.

Mathematical Definition

For a continuous function f(x) defined on the interval [a, b], the area under the curve from x = a to x = b is given by the definite integral: $Area = \int_{a}^{b} f (x) d x$

1.0Area Under the Curve Definition

The area under the curve is a fundamental concept in integral calculus that quantifies the total area between the graph of a function and the x-axis over a specified interval. Mathematically, it is determined using definite integrals.

Mathematical Representation:

If f(x) is a continuous function defined on the interval [a, b], the area under the curve from x = a to x = b is given by the definite integral: $Area = \int_{a}^{b} f (x) d x$

2.0Area Under the Curve – Between a Curve and Coordinate Axis

Area under the Curve and x- axis (Vertical Strips)

a.

Area of the strip = y.dx

Area bounded by the curve, the x-axis and the ordinate at x = a and x = b is given by A= $\int_{a}^{b} y d x, where y = f (x)$ lies above the x-axis and b > a.

Here vertical strip of thickness dx is considered at distance x.

b.

If y = f(x) lies completely below the x-axis, then $A = \int_{a}^{b} y d x$

c.

If curve crosses the $x_{-axis at} x = c, then A = \int_{a}^{c} y d x + \int_{c}^{b} y d x$

Area under the Curve and y- axis (Horizontal Strips)

a.

Area of the strip = x.dy

Graph of x = g(y)

If g(y) $\geq 0 for y \in [c, d]$ then area bounded by curve x = g(y) and y-axis between abscissa y = c and y = d is $\int_{y = c}^{d} g (y) d y$

b. If g(y) $\leq 0 for y \in [c, d]$ then area bounded by curve x = g(y) and y-axis between abscissa y = c and y = d is - $\int_{y = c}^{d} g (y) d y$

Note:

General formula for area bounded by curve x = g(y) and y-axis between abscissa y = c and y=d $is \int_{y = c}^{d} ∣ g (y) ∣ d y$ .

3.0Area under the curve - Symmetric Area

If the curve is symmetric in all four quadrants, then

Total area = 4 (Area in any one of the quadrants).

4.0Area Under a curve - Between Two Curves

Area bounded by two curves y=f(x) \& y=g(x)

such that f(x)>g(x) is

$A = \int_{x_{1}}^{x_{2}} [f (x) - g (x)] d x$ where x₁ and x₂ are roots of equation f(x) = g (x)

In case horizontal strip is taken we have

$A = \int_{y_{1}}^{y_{2}} [f (y) - g (y)] d y$

Where y₁ & y₂ are roots of equation f(y) = g(y)

If the curves y = f(x) and y = g(x) intersect at x = c , then required area $A = \int_{a}^{c} (g (x) - f (x)) d x + \int_{c}^{b} (f (x) - g (x)) d x = \int_{a}^{b} ∣ f (x) - g (x) ∣ d x$

Note: Required area must have all the boundaries indicated in the problem.

5.0Standard Areas (To be Remembered)

Area bounded by parabolas $y^{2} = 4 a x; x^{2} = 4 b y, a > 0; b > 0 i s \frac{16 ab}{3}$

Intersection point:

$x^{2} = 4 b y y^{2} = 4 a x}$

$x^{2} = 4 b 4 a x$

$x^{4} = 64 b^{2} a x$

x = 0 or $x = 4 b^{2/3} a^{1/3}$

$A = \int_{0}^{4 b^{2/3/ a^{2/3}}} (4 a x - \frac{x ^{2}}{4 b}) d x = \frac{16 ab}{3}$

Whole area of the ellipse, $x^{2} / a^{2} + y^{2} / b^{2} = 1$ is $πab$ sq. units.
Area included between the parabola $y^{2} = 4 a x$ & the line y = mx is $8 a^{2} /3 m^{3}$ sq. units.
The area of the region bounded by one arch of sin ax (or cos ax) and x-axis is $2/ a$ sq. units.
Average value of a function y = f(x) over an interval a $\leq x \leq b$ is defined as:

$y (a v) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

If y = f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at x = a, x= b, y = f(x) and y = f(c) $[w h ere c_{\in} (a, b)]$ is minimum when $c = \frac{a + b}{2}$ .

6.0Solved Examples on Area Under the Curve

Example 1: Find area bounded by $x = 1, x = 2, y = x^{2}, y = 0$ .

Solution:

$A = \int_{1}^{2} (x^{2} - 0) d x$

$= \frac{[ x ^{3} ] _{1}^{2}}{3}$

$= \frac{8 - 1}{3} = \frac{7}{3}$

Example 2: Find the area in the first quadrant bounded by $y = 4 x^{2}, x = 0, y = 1 and y = 4$ .

Solution:

Required area = $\int_{1}^{4} x d y = \int_{1}^{4} \frac{y}{2} d y$

$= \frac{1}{2} \cdot \frac{2}{3} \cdot (y^{\frac{3}{2}})_{1}^{4}$

$= \frac{1}{3} [4^{\frac{3}{2}} - 1]$

$= \frac{1}{3} [8 - 1] = \frac{7}{3} = 2 \frac{1}{3} sq. units.$

Example 3: Find the area enclosed between $y = sin x; y = cos x$ and y-axis in the 1^stquadrant

Solution:

$A = \int_{0}^{\frac{π}{4}} (cos x - sin x) d x$

$= [sin x + cos x]_{0}^{π /4} = (\frac{1}{2} + \frac{1}{2}) - (0 + 1)$

$= 2 - 1$

Example 4: Find the area bounded by the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$

Solution:

Area bounded by ellipse in first quadrant = $\int_{0}^{3} \frac{2}{3} 9 - x^{2} d x = \frac{3 π}{2}$

∵ Curve is symmetrical about all four quadrants

∴ Total area = 4 (Area in any one of the quadrants)

$= 4 (\frac{3 π}{2}) = 6 π$

Example 5: Compute the area of the figure bounded by the parabolas $x = - 2 y^{2}, x = 1 - 3 y^{2} .$

Solution:

Solving the equations $x = - 2 y^{2}, x = 1 - 3 y^{2}$ ,

we find that ordinates of the points of intersection

of the two curves as $y_{1} = - 1, y_{2} = 1$

The points are (–2, –1) and (–2, 1).

The required area

$2 \int_{0}^{1} (x_{1} - x_{2}) d y = 2 \int_{0}^{1} [(1 - 3 y^{2}) - (- 2 y^{2})] d y = 2 \int_{0}^{1} (1 - y^{2}) d y$

$= 2 [y - \frac{y ^{3}}{3}]_{0}^{1} = \frac{4}{3} sq.units.$

7.0Practice Problems on Area Under the Curve

1. Find the area bounded by y = x² + 2 above x-axis between x = 2 and x = 3.

2. Find the area bounded by the curve y = cos x and the x-axis from x = 0 to x = 2π.

3. Find the area bounded by x = 2, x = 5, y = x², y = 0.

4. Find the area bounded by y = x² and y = x.

5. A figure is bounded by the curvesy =\left|\sqrt{2} \sin \frac{\pi x}{4}\right|, y = 0, x = 2 and x = 4. At what angles to the positive x-axis straight lines must be drawn through (4,0) so that these lines partition the figure into three parts of the same area.

8.0Solved Questions on Area Under the Curve

How to Find Area Under the Curve?

Ans: The area under the curve is calculated using definite integrals. The integral of a function f(x) from a to b is given by:

$Area = \int_{a}^{b} f (x) d x$

This integral represents the net area between the curve and the x-axis over the interval [a, b].

What is the Formula for the Area Under the Curve?

Ans: The formula to find the area under the curve is:

$Area = \int_{a}^{b} f (x) d x$

where f(x) is the function describing the curve, and a and b are the lower and upper limits of the interval.

Frequently Asked Questions

Join ALLEN!

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Class

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Preferred Programs

Preferred Mode

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I agree to

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Area Under the Curve: Definition and Importance

The area under the curve is a concept in integral calculus that quantifies the total area enclosed by a curve, the x-axis, and specified vertical boundaries on the graph of a function. This area is calculated using definite integrals and represents the accumulation of a quantity described by the function over a given interval.

Mathematical Definition

For a continuous function f(x) defined on the interval [a, b], the area under the curve from x = a to x = b is given by the definite integral: $Area = \int_{a}^{b} f (x) d x$

1.0Area Under the Curve Definition

The area under the curve is a fundamental concept in integral calculus that quantifies the total area between the graph of a function and the x-axis over a specified interval. Mathematically, it is determined using definite integrals.

Mathematical Representation:

If f(x) is a continuous function defined on the interval [a, b], the area under the curve from x = a to x = b is given by the definite integral: $Area = \int_{a}^{b} f (x) d x$

2.0Area Under the Curve – Between a Curve and Coordinate Axis

Area under the Curve and x- axis (Vertical Strips)

a.

Area of the strip = y.dx

Area bounded by the curve, the x-axis and the ordinate at x = a and x = b is given by A= $\int_{a}^{b} y d x, where y = f (x)$ lies above the x-axis and b > a.

Here vertical strip of thickness dx is considered at distance x.

b.

If y = f(x) lies completely below the x-axis, then $A = \int_{a}^{b} y d x$

c.

If curve crosses the $x_{-axis at} x = c, then A = \int_{a}^{c} y d x + \int_{c}^{b} y d x$

Area under the Curve and y- axis (Horizontal Strips)

a.

Area of the strip = x.dy

Graph of x = g(y)

If g(y) $\geq 0 for y \in [c, d]$ then area bounded by curve x = g(y) and y-axis between abscissa y = c and y = d is $\int_{y = c}^{d} g (y) d y$

b. If g(y) $\leq 0 for y \in [c, d]$ then area bounded by curve x = g(y) and y-axis between abscissa y = c and y = d is - $\int_{y = c}^{d} g (y) d y$

Note:

General formula for area bounded by curve x = g(y) and y-axis between abscissa y = c and y=d $is \int_{y = c}^{d} ∣ g (y) ∣ d y$ .

3.0Area under the curve - Symmetric Area

If the curve is symmetric in all four quadrants, then

Total area = 4 (Area in any one of the quadrants).

4.0Area Under a curve - Between Two Curves

Area bounded by two curves y=f(x) \& y=g(x)

such that f(x)>g(x) is

$A = \int_{x_{1}}^{x_{2}} [f (x) - g (x)] d x$ where x₁ and x₂ are roots of equation f(x) = g (x)

In case horizontal strip is taken we have

$A = \int_{y_{1}}^{y_{2}} [f (y) - g (y)] d y$

Where y₁ & y₂ are roots of equation f(y) = g(y)

If the curves y = f(x) and y = g(x) intersect at x = c , then required area $A = \int_{a}^{c} (g (x) - f (x)) d x + \int_{c}^{b} (f (x) - g (x)) d x = \int_{a}^{b} ∣ f (x) - g (x) ∣ d x$

Note: Required area must have all the boundaries indicated in the problem.

5.0Standard Areas (To be Remembered)

Area bounded by parabolas $y^{2} = 4 a x; x^{2} = 4 b y, a > 0; b > 0 i s \frac{16 ab}{3}$

Intersection point:

$x^{2} = 4 b y y^{2} = 4 a x}$

$x^{2} = 4 b 4 a x$

$x^{4} = 64 b^{2} a x$

x = 0 or $x = 4 b^{2/3} a^{1/3}$

$A = \int_{0}^{4 b^{2/3/ a^{2/3}}} (4 a x - \frac{x ^{2}}{4 b}) d x = \frac{16 ab}{3}$

Whole area of the ellipse, $x^{2} / a^{2} + y^{2} / b^{2} = 1$ is $πab$ sq. units.
Area included between the parabola $y^{2} = 4 a x$ & the line y = mx is $8 a^{2} /3 m^{3}$ sq. units.
The area of the region bounded by one arch of sin ax (or cos ax) and x-axis is $2/ a$ sq. units.
Average value of a function y = f(x) over an interval a $\leq x \leq b$ is defined as:

$y (a v) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

If y = f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at x = a, x= b, y = f(x) and y = f(c) $[w h ere c_{\in} (a, b)]$ is minimum when $c = \frac{a + b}{2}$ .

6.0Solved Examples on Area Under the Curve

Example 1: Find area bounded by $x = 1, x = 2, y = x^{2}, y = 0$ .

Solution:

$A = \int_{1}^{2} (x^{2} - 0) d x$

$= \frac{[ x ^{3} ] _{1}^{2}}{3}$

$= \frac{8 - 1}{3} = \frac{7}{3}$

Example 2: Find the area in the first quadrant bounded by $y = 4 x^{2}, x = 0, y = 1 and y = 4$ .

Solution:

Required area = $\int_{1}^{4} x d y = \int_{1}^{4} \frac{y}{2} d y$

$= \frac{1}{2} \cdot \frac{2}{3} \cdot (y^{\frac{3}{2}})_{1}^{4}$

$= \frac{1}{3} [4^{\frac{3}{2}} - 1]$

$= \frac{1}{3} [8 - 1] = \frac{7}{3} = 2 \frac{1}{3} sq. units.$

Example 3: Find the area enclosed between $y = sin x; y = cos x$ and y-axis in the 1^stquadrant

Solution:

$A = \int_{0}^{\frac{π}{4}} (cos x - sin x) d x$

$= [sin x + cos x]_{0}^{π /4} = (\frac{1}{2} + \frac{1}{2}) - (0 + 1)$

$= 2 - 1$

Example 4: Find the area bounded by the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$

Solution:

Area bounded by ellipse in first quadrant = $\int_{0}^{3} \frac{2}{3} 9 - x^{2} d x = \frac{3 π}{2}$

∵ Curve is symmetrical about all four quadrants

∴ Total area = 4 (Area in any one of the quadrants)

$= 4 (\frac{3 π}{2}) = 6 π$

Example 5: Compute the area of the figure bounded by the parabolas $x = - 2 y^{2}, x = 1 - 3 y^{2} .$

Solution:

Solving the equations $x = - 2 y^{2}, x = 1 - 3 y^{2}$ ,

we find that ordinates of the points of intersection

of the two curves as $y_{1} = - 1, y_{2} = 1$

The points are (–2, –1) and (–2, 1).

The required area

$2 \int_{0}^{1} (x_{1} - x_{2}) d y = 2 \int_{0}^{1} [(1 - 3 y^{2}) - (- 2 y^{2})] d y = 2 \int_{0}^{1} (1 - y^{2}) d y$

$= 2 [y - \frac{y ^{3}}{3}]_{0}^{1} = \frac{4}{3} sq.units.$

7.0Practice Problems on Area Under the Curve

1. Find the area bounded by y = x² + 2 above x-axis between x = 2 and x = 3.

2. Find the area bounded by the curve y = cos x and the x-axis from x = 0 to x = 2π.

3. Find the area bounded by x = 2, x = 5, y = x², y = 0.

4. Find the area bounded by y = x² and y = x.

5. A figure is bounded by the curvesy =\left|\sqrt{2} \sin \frac{\pi x}{4}\right|, y = 0, x = 2 and x = 4. At what angles to the positive x-axis straight lines must be drawn through (4,0) so that these lines partition the figure into three parts of the same area.

8.0Solved Questions on Area Under the Curve

How to Find Area Under the Curve?

Ans: The area under the curve is calculated using definite integrals. The integral of a function f(x) from a to b is given by:

$Area = \int_{a}^{b} f (x) d x$

This integral represents the net area between the curve and the x-axis over the interval [a, b].

What is the Formula for the Area Under the Curve?

Ans: The formula to find the area under the curve is:

$Area = \int_{a}^{b} f (x) d x$

where f(x) is the function describing the curve, and a and b are the lower and upper limits of the interval.

Frequently Asked Questions

What is the Area Under the Curve?

What Methods Can Be Used to Find the Area Under the Curve?

How Do You Find the Area Between a Curve and a Line?

How Do You Calculate the Area Between Two Curves?

Join ALLEN!

Related Articles:-

Area of Sector

Area of Ellipse

Area of Hollow Cylinder

Logarithm

Binomial Theorem

Area Under the Curve: Definition and Importance

1.0Area Under the Curve Definition

2.0Area Under the Curve – Between a Curve and Coordinate Axis

Area under the Curve and x- axis (Vertical Strips)

Area under the Curve and y- axis (Horizontal Strips)

3.0Area under the curve - Symmetric Area

4.0Area Under a curve - Between Two Curves

5.0Standard Areas (To be Remembered)

6.0Solved Examples on Area Under the Curve

7.0Practice Problems on Area Under the Curve

8.0Solved Questions on Area Under the Curve

Table of Contents

Frequently Asked Questions

What is the Area Under the Curve?

What Methods Can Be Used to Find the Area Under the Curve?

How Do You Find the Area Between a Curve and a Line?

How Do You Calculate the Area Between Two Curves?

Join ALLEN!

Related Articles:-

Area of Sector

Area of Ellipse

Area of Hollow Cylinder

Logarithm

Binomial Theorem

Area Under the Curve: Definition and Importance

1.0Area Under the Curve Definition

2.0Area Under the Curve – Between a Curve and Coordinate Axis

Area under the Curve and x- axis (Vertical Strips)

Area under the Curve and y- axis (Horizontal Strips)

3.0Area under the curve - Symmetric Area

4.0Area Under a curve - Between Two Curves

5.0Standard Areas (To be Remembered)

6.0Solved Examples on Area Under the Curve

7.0Practice Problems on Area Under the Curve

8.0Solved Questions on Area Under the Curve

Table of Contents