 # 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:

Area= $\int_{a}^{b}ydx=\int_{a}^{b}f(x)dx$

If curvy region is x=$\phi (y)$, y-axis and the line y=c, y=d is represented through this formula:

Area=$\int_{c}^{d}xdy=\int_{c}^{d}\phi (y)dy$

### 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:

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

If f(x)$\geq$g(x) in [a,c] and $f(x)\leq g(x)$ in [c,b], a<c<b, then area is

$\int_{a}^{c}[f(x)-g(x)]dx+\int_{c}^{b}[g(x)-f(x)]dx$

### 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

1. 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
2. Find the area bounded by curves {(x, y) : y ≥ $x^{2}$ and y = | x |}
3. Find the area of the region enclosed by the parabola $x^{2}$ = y, the line y = x+ 2 and the x-axis
4. Using integration find the area of a region bounded by the triangle whose vertices
are (– 2, 1), (0, 4) and (4, 3).