How to Find the Area of the Shaded Region Calculus
Area Between Two Curves
Finding the area between two curves is an essential application of integration. By using integration, we have learned to find the area under the curve, similarly, we can also find the area between two intersecting curves using integration. It is the portion of the space that falls between two linear or non-linear curves within the given limits.
The area between two curves can be composite also, but by using integration we can also find that easily by doing simple modifications to the known formulas used for finding the area under two curves. Let us discuss the topic in the following content.
| 1. | Area Between Two Curves Introduction |
| 2. | Area Between Two Curves Formula |
| 3. | Area Between Two Curves With Respect to Y |
| 4. | Area Between Two Compound Curves |
| 5. | Area Between Two Polar Curves |
| 6. | Area Between Two Curves Examples |
| 7. | Practice Questions |
| 8. | Frequently Asked Questions on Area Between Two Curves |
Area Between Two Curves Introduction
The area between two curves is the area that falls in between two intersecting curves and can be calculated using integral calculus. Integration can be used to find the area under two curves where we know the equation of two curves, and their intersection points. If we see in the image, we have two functions f(x) and g(x) and we need to find the area between these two curves given in the shaded portion. Then by using integration, we can easily calculate the area of the shaded portion. Let us discuss more on the calculation of this area in the next section.
Area Between Two Curves Formula
If we want to find the area between two curves, we need to divide the area into many small rectangular strips parallel to the y-axis, starting from x = a to x = b, and by using integration we can add the areas of these small strips to get the approximation area of two curves. These rectangular strips will have the width "dx" and height f(x) - g(x). The area of the small rectangular strip is dx(f(x) - g(x)) and now by using integration within the limits x = a and x = b, we can calculate the area between these two curves. If f(x) and g(x) are continuous on [a, b] and g(x) < f(x) for all x in [a, b], then we have the following formula.
Area = \(\int_{a}^{b} \left [ f(x) - g(x) \right ] \;dx\)
Area Between Two Curves With Respect to Y
The area between two curves with respect to the y-axis is the method of calculating the areas of the curves whose equation is given in terms of y. Calculating the area along the y-axis is easier compared to calculating the area along the x-axis. In this method, we divide the given region into horizontal strips between the given limits, and by using integration, we add the areas of the horizontal strips to find the area of the section between two curves. If f(y) and g(y) are continuous on [c, d] and g(y) < f(y) for all y in [c, d], then
Area = \(\int_{a}^{b} \left [ f(y) - g(y) \right ] \;dy\)
Area Between Two Compound Curves
Calculating areas between two compound curve which intersect with each other using above stated formulas will give the incorrect result and curves change places after the intersection. For the curves shown in the image, we divided the intervals into various portions and then calculate individual areas between the curves in each section. Let f(x) and g(x) be continuous in [a,b] interval, the area between the curves will be:
Area = \(\int ^c_a|f(x)−g(x)|dx\)
As we see in the region [a, b], f(x) ≥ g(x) and in the region [c, d] g(x) ≥ f(x), so we break the limits into two parts as:
Area = \(\int ^b_a(f(x)−g(x))dx + \int ^c_b(g(x)−f(x))dx\)
Area Between Two Polar Curves
By using integral calculus we can calculate the area between two polar curves as well. When we have two curves whose coordinates are not given in rectangular coordinates, but in polar coordinates, we use this method. We can always convert the polar to rectangular coordinates also to solve this, but we can use this method to reduce the complexity. Let us say we have two polar curves \( r_0\) = f(θ) and \( r_i\) = 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. Then the area between the curves will be:
\( A = \dfrac{1}{2}\int ^β_α(r^2_0- r^2_i) dθ \) -
Area Between Two Curves Examples
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Example 1: Find the area between two curves f(x) = x2 and g(x) = x3 within the interval [0,1] where f(x) ≥ g(x) in the given region.
Solution:
Given: f(x) = x2 and g(x) = x3
Using the formula for the area between two curves:
Area = \(\int_{a}^{b} \left [ f(x) - g(x) \right ] \;dx\)
Area = \( \int_{0}^{1} [ x^2 - x^3 ] dx\)
= \( {\left[ \frac{1}{3}x^3 - \frac{1}{4}x^4 \right]}_0^1 \)
= 1/3 - 1/4
= 1/12
Answer: The area between the given curves under the following interval is 1/12 unit square.
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Example 2: Find the area between the curve \(r_0\) = 2cos(θ) and \(r_i\)= 1 in the interval [−π/3, π/3] and where −π/3 < θ < π/3 and in the given region \(r_0\) ≥ \(r_i\).
Solution:
Given : \(r_0\) = 2cos(θ) and \(r_i\)= 1
Using the formula for the area between two polar curves:
\( A = \dfrac{1}{2}\int ^β_α(r^2_0- r^2_i) dθ \)
\( A = \dfrac{1}{2}\int ^{π/3}_{−π/3}((2cos \theta)^2- (1)^2) dθ \)
A = \(\frac12 \int_{-\pi/3}^{\pi/3} 1 + 2 \cos(2\theta) \, d\theta \)
= \( \frac12 (\theta + \sin(2\theta)) \, \Big |_{-\pi/3}^{\pi/3} \)
= π/3+√3/2.
Answer: The area between the curve is π/3+√3/2 square units.
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Practice Questions
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Frequently Asked Questions on Area Between Two Curves
What Does Area Under the Curve Mean?
The area under the curve means the area bounded by the curve, the axis, and the boundary points. The area under the curve is a two-dimensional area, which can be calculated with the help of the coordinate axes and by using the integration formula.
What Does Area Under the Curve Represent?
The area under the curve represents the area enclosed under the curve and the axis, which is marked with limiting points. This area under the curve gives the area of the irregular plane shape in a two-dimensional array.
What is the Formula for Area Between Two Curves?
We can calculate the area between these two curves by using the given formula. If f(y) and g(y) are continuous on [a, b] and g(y) f(y) for all y in [a, b], then
Area = \(\int_{a}^{b} \left [ f(x) - g(x) \right ] \;dx\)
Is the Area Between Curves Always Positive?
Yes, if there exists the area between two curves, then it will always be a non-negative value. The area can be 0 or any positive value, but it can never be negative. The area between two curves is calculated by the formula: Area = \(\int_{a}^{b} \left [ f(x) - g(x) \right ] \;dx\) which is an absolute value of the area. It can never be negative.
How to Find the Area Under the Curve?
The area under the curve can be found using the process of integration or antiderivative. For this, we need the equation of the curve(y = f(x)), the axis bounding the curve, and the boundary limits of the curve. With this the area bounded under the curve can be calculated with the formula A = \(\int_a^by.dx\).
What Are the Different Methods to Find the Area Under the Curve?
There are three broad methods to find the area under the curve. The area under the curve is calculated by dividing the area space into numerous small rectangles, and then the areas are added to obtain the total area. The second method is to divide the area into a few rectangles and then the areas are added to obtain the required area. The third method is to find the area with the help of integration.
How to Find the Area of the Shaded Region Calculus
Source: https://www.cuemath.com/calculus/area-between-two-curves/