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).

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