Application of Integrals Class 12 Notes Chapter 8

CBSE Class 12 Maths Notes Chapter 8 Application of Integrals – Related Links

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:

Area=

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

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:

Area=

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

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

\(\begin{array}{l}\int_{a}^{c}[f(x)-g(x)]dx+\int_{c}^{b}[g(x)-f(x)]dx\end{array} \)

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

(B)1/3

(C)2/3

(D)4/3

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:

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

What is Calculus’?

Branch of mathematics 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 building.

What are the uses of ‘Differential calculus’?

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

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