By going through these CBSE Class 12 Maths Notes Chapter 8 Application of Integrals, students can recall all the concepts quickly.

## Application of Integrals Notes Class 12 Maths Chapter 8

**Area under Simple curves:**

1. Let us find the area bounded by the curve y = f(x), x-axis, and the ordinates x = a and x – b. Consider the area under the curve as composed of a large number of thin vertical stripes.

Let there be an arbitrary strip of height y and width dx.

Area of elementary strip dA = y dx, where y = f(x).

Total area A of the region between x-axis, ordinates x – a, x = b and the curve y = f(x)

= sum of areas of elementary thin strips across the region PQML.

A = ∫_{a}^{b} dA = ∫_{a}^{b} ydx = ∫_{a}^{b} f(x) dx.

2. The area A of the region bounded by the curve x = g(y), y-axis, and the lines

y = c and y = d is given by

A = ∫_{c}^{d} x dy

3. If the curve under consideration lies below x-axis, then f(x) < 0 from x = a to x = b. So, the area bounded by the curve y = f(x) and the ordinates x = a, x = b and x-axis is negative. But the numerical value of the area is to be taken into consideration.

Then, area = |∫_{a}^{b} f(x)dx|.

4. It may also happen that some portion of the curve is above the x-axis and some portion is below the x-axis as shown in the figure. Let A_{1} be the area below the x-axis and A_{2} be the area above the x-axis. Therefore, area A bounded by the curve y = f(x), x-axis and the ordinates x = a and x = b is given by

A = |A_{1}| + A_{2}.

**Area between two curves:**

1. Let the two curves by y = f(x) and y = g(x), as shown in the figure. Suppose these curve intersect at x = a and x = b.

Consider the elementary strip of height y where y = f(x) – g(x), with width dx.

∴ dA = y dx.

⇒ A = ∫_{a}^{b} (f(x) – g(x))dx

= ∫_{a}^{b} f(x) dx – ∫_{a}^{b} g(x) dx.

= Area bounded by the curve y = f(x) – Area bounded by the curve y = g(x), where f(x) > g(x).

2. If the two curves y = f(x) and y = g(x) intersect at x-a,x – c and x = b such that a < c < b, then:

If f(x) > g(x) in [a, c] and f(x) < g(x) in [c, b], then the area of the regions bounded curve

= Area of the region PAQCP + Area of the region QDRBQ

= ∫_{a}^{c} f(x) – g(x)) dx + ∫_{c}^{b} (g(x) – f(x)) dx.

1. Area Under Simple Curves

(i) Area of the region bounded by the curve y = f (x), x-axis and the linesx = a and x = b(b > a) is given by the formula:

Area = \(\int_{a}^{b}\) ydy = \(\int_{a}^{b}\) f(x) dy.

2. Area of the region bounded by the curve x = g(x), y-axis and the lines y = c,y = d is given by the formula:

Area = \(\int_{c}^{d}\) xdy = \(\int_{c}^{d}\) g(y) dy.

2. Area Between two Curves

(i) Area of the region enclosed between two curves y = f (x), y = g (x) and the lines x = a, x = b is

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

(ii) Iff (x) ≥ g (x) in [a, c] and f(x) ≤ g (x) in [c, b], a < c < b, then we write the area as:

Area = \(\int_{a}^{c}\) [f(x) – g(x)] dx + \(\int_{c}^{b}\) [g(x) – f(x)] dx.