Maharashtra Board Class 12 Maths Commerce Part I Chapter 7 Application of Definite Integration PDF Download

Applications of Definite Integration

Introduction

The theory of integration has a large variety of applications in Science and Engineering. In this chapter we shall use integration for finding the area of a bounded region. For this, we first draw the sketch (if possible) of the curve which encloses the region. For evaluation of area bounded by the certain curves, we need to know the nature of the curves and their graphs.

The shapes of different types of curves are discussed below.

7.1 Standard Forms Of Parabola And Their Shapes

1. \(y^2 = 4ax\)

2. \(y^2 = -4ax\)

3. \(x^2 = 4by\)

4. \(x^2 = -4by\)

Teacher's Note

A parabola is a curve that looks like the shape of a ball thrown in the air. You can see parabolas in the path of water coming out of a tube well in a field.

Exam Trick

Remember: If \(y^2\) is given, the parabola opens left or right. If \(x^2\) is given, the parabola opens up or down.

Points to Remember

Parabola with \(y^2 = 4ax\) opens to the right.
Parabola with \(y^2 = -4ax\) opens to the left.
Parabola with \(x^2 = 4by\) opens upward.
Parabola with \(x^2 = -4by\) opens downward.

7.2 Standard Forms Of Ellipse

1) \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (where \(a > b\))

2) \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (where \(a < b\))

Teacher's Note

An ellipse is an oval shape. You can see ellipses in the orbits of planets around the sun, or in the shape of an egg.

Exam Trick

Remember: In an ellipse, \(a\) and \(b\) are the lengths from the center to the edge. The bigger number tells you which way the ellipse is longer.

Points to Remember

An ellipse is an oval curve with a center point.
The value \(a\) is the distance from center to the edge along the x-axis.
The value \(b\) is the distance from center to the edge along the y-axis.
If \(a > b\), the ellipse is wider than it is tall.

7.3 Area Under The Curve

To find the area under the curve, we state only formulae without proof.

(1) The area "A" bounded by the curve \(y = f(x)\), X-axis and bounded between the lines \(x = a\) and \(x = b\) is given by

\[A = \text{Area of the region PRSQ} = \int_a^b y \, dx = \int_a^b f(x) \, dx\]

(2) The area A bounded by the curve \(x = g(y)\), Y-axis and bounded between the lines \(y = c\) and \(y = d\) is given by

\[A = \int_c^d x \, dy = \int_c^d g(y) \, dy\]

(3) The area of the shaded region bounded by two curves \(y = f(x)\), \(y = g(x)\) is obtained by

\[A = \left|\int_{x=a}^{x=b} f(x) \, dx - \int_{x=a}^{x=b} g(x) \, dx\right|\]

where the curve \(y = f(x)\) and \(y = g(x)\) intersect at points \((a, f(a))\) and \((b, f(b))\).

Remarks

(i) If the curve under consideration is below the X-axis, then the area bounded by the curve, X-axis and lines \(x = a\), \(x = b\) is negative.

We consider the absolute value in this case.

Thus, required area = \(\left|\int_{x=a}^{x=b} f(x) \, dx\right|\)

Teacher's Note

Area means we always count it as a positive number. Even if the curve is below the line, we take the absolute value so the area is never negative.

Exam Trick

Remember: Area below the x-axis is still positive! We use the absolute value (remove the minus sign) to make it positive.

Points to Remember

Area is always a positive number.
If the curve is below the x-axis, take absolute value.
The area between two curves is the difference of their integrals.
Always put limits of integration in the integral sign.

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