Application of Integrals Class 12 Notes Chapter 8

Integrals used to find the area of a curved region

Integrals are the functions which satisfy a given differential equation for finding the area of a curvy region y=f(x), the x-axis and the line x=a and x=b(b>a) is represented through this formula:


\(\begin{array}{l}\int_{a}^{b}ydx=\int_{a}^{b}f(x)dx\end{array} \)

If curvy region is x=

\(\begin{array}{l}\phi (y)\end{array} \)
, y-axis and the line y=c, y=d is represented through this formula:


\(\begin{array}{l}\int_{c}^{d}xdy=\int_{c}^{d}\phi (y)dy\end{array} \)

Area Bounded By Two Curves

If the dimensions of two curves are y=f(x), y=g(x) and lines x=a and x=b is represented by the formula:

\(\begin{array}{l}\int_{a}^{b}[f(x)-g(x)]dx\end{array} \)
, where f(x)
\(\begin{array}{l}\geq\end{array} \)
g(x) in [a,b]

If f(x)

\(\begin{array}{l}\geq\end{array} \)
g(x) in [a,c] and
\(\begin{array}{l}f(x)\leq g(x)\end{array} \)
in [c,b], a<c<b, then area is
\(\begin{array}{l}\int_{a}^{c}[f(x)-g(x)]dx+\int_{c}^{b}[g(x)-f(x)]dx\end{array} \)
Also Access 
NCERT Solutions for Class 12 Maths Chapter 8
NCERT Exemplar for Class 12 Maths Chapter 8

Important Questions:

1. The area bounded by the curve p = y | y | , x-axis and the ordinates y = – 1 and y = 1 is given by

(A) 0




2. Using the method of integration, find the area of the region bounded by lines:
3y + z = 5, 4y – z = 6 and y – 4z + 6 = 0

3. Find the area bounded by curves {(x, y) : y ≥

\(\begin{array}{l}x^{2}\end{array} \)
and y = | x |}

4. Find the area of the region enclosed by the parabola

\(\begin{array}{l}x^{2}\end{array} \)
= y, the line y = x+ 2 and the x-axis

5. Using integration, find the area of a region bounded by the triangle whose vertices are (– 2, 1), (0, 4) and (4, 3).

Also, Read

Integral Calculus Indefinite Integrals

Frequently Asked Questions on CBSE Class 12 Maths Notes Chapter 8 Application of Integrals


What is calculus?

The branch of mathematics is concerned with the calculation of instantaneous rates of change (differential calculus) and the summation of infinitely many small factors to determine some whole (integral calculus).


What are the real-life uses of integrals?

In real life, integrations are used in various fields, such as engineering, where engineers use integrals to find the shape of the building.


What are the uses of differential calculus?

1. Statistics 2. Data evaluation 3. Credit card/debit card calculations 4. Space technology 5. Mechanical/civil engineering industries


Leave a Comment

Your Mobile number and Email id will not be published.



Tuition Center
Tuition Centre
free trial
Free Trial Class
Question and Answer
Question & Answer