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Area Between Two Curves - Calculus - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Area Bounded by Two Curves is considered one the most difficult concept.

  • Area Bounded by Curve and Axes is considered one of the most asked concept.

  • 115 Questions around this concept.

Solve by difficulty

The area enclosed between the curves y^{2}=x\; and\; y=\left | x \right |\; is

Concepts Covered - 3

Area Bounded by Curve and Axes

Area Bounded by Curve and Axes

In the previous concept we learned that if the function  f(x) ≥ 0 ∀ x ∈ [a, b] then \int^{b}_af(x)\;dx represents the area bounded by y = f(x), x-axis and lines x = a and x = b.

If the function  f(x) ≤ 0 ∀ x ∈ [a, b],  then the area by bounded y = f(x), x-axis and lines x = a and x = b is \left |\int^{b}_af(x)\;dx \right |.

 

 

Area along Y-axis

The area by bounded x = g(y) [with g(y)>0], y-axis and the lines y = a and y = b is \int^{b}_ax\;dy=\int^{b}_ag(y)\;dy

 

Area of Piecewise Function

If the graph of the function f(x) is of the following form, then

 

\\\text {then } \int_{a}^{b} f(x) d x \text { will equal } A_{1}-A_{2}+A_{3}-A_{4} \text { and not }A_{1}+A_{2}+A_{3}+A_{4}. \\\\\text {If we need to evaluate } A_{1}+A_{2}+A_{3}+A_{4} \text { (the magnitude of the bounded area),} \\\text{we will have to calculate}

\underbrace{\int_{a}^{x}f(x)\;dx}\;\;+\;\;\underbrace{\left | \int_{x}^{y}f(x)\;dx \right |}\;\;+\;\;\underbrace{\int_{y}^{z}f(x)\;dx} \;\;+\;\;\underbrace{\left | \int_{z}^{b}f(x)\;dx \right |}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;A_1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;A_2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;A_3\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;A_4}

 

The area bounded by curve when curve intersects X-axis

The graph y = f(x) ∀  x ∈ [a, b] intersects x-axis at x = c.

If the function  f(x) ≥ 0 ∀ x ∈ [a, c] and f(x) ≤ 0 ∀ x ∈ [c, b] then area bounded by curve and x-axis, between lines x = a and x = b is 

\int_{a}^{b}\left |f(x) \right |\;dx=\int_{a}^{c} f(x) \;dx-\int_{c}^{b} f(x) \;dx

Area Bounded by Two Curves

Area bounded by the curves  y=f(x),  y=g(x)  and the lines  x = a and  x = b, and it is given that f(x) ≤ g(x).

 

From the figure, it is clear that, 

Area of the shaded region = Area of the region ABEF -  Area of the region ABCD

 

\int_{a}^{b}g(x)\;dx-\int_{a}^{b}f(x)\;dx=\int_{a}^{b}\left (\underbrace{\;\;\;g(x)\;\;}-\underbrace{\;\;\;f(x)\;\;} \right )\;dx\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\text{upper \;\;\;\;\;\;\;lower}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\text{curve \;\;\;\;\;\;\;curve}

Area Bounded by Curves When Intersects at More Than One Point

Area bounded by the curves  y = f(x),  y = g(x) which intersect each other in the interval [a, b]

First find the point of intersection of these curves  y = f(x) and  y = g(x) by solving the equation f(x) = g(x), let the point of intersection be x = c

Area of the shaded region  

=\int_{a}^{c}\{f(x)-g(x)\} d x+\int_{c}^{b}\{g(x)-f(x)\} d x

 

When two curves intersects more than one point

Area bounded by the curves  y=f(x),  y=g(x) which intersect each other at three points at  x = a, x = b and x = c.

To find the point of intersection, solve f(x) = g(x).

For x ∈ (a, c), f(x) > g(x) and for x ∈ (c, b), g(x) > f(x).

Area bounded by curves,

\\\mathrm{A=} \int_{a}^{b}\left |f(x)-g(x) \right |dx\\\\\mathrm{\;\;\;\;=} \int_{a}^{c}\left ( f(x)-g(x) \right )dx+\int_{c}^{b}\left ( g(x)-f(x) \right )dx  

 

Study it with Videos

Area Bounded by Curve and Axes
Area Bounded by Two Curves
Area Bounded by Curves When Intersects at More Than One Point

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Books

Reference Books

Area Bounded by Two Curves

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : *.6

Line : 41

Area Bounded by Curves When Intersects at More Than One Point

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 9.10

Line : 25

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