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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

Syllabus

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.
110 Questions around this concept.

Solve by difficulty

EASY

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

1/6

1/3

2/3

View Solution

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,

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