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Maths: Integral Calculus: The area of the region bounded by the curves: Geometrical Interpretation of Definite Integral as Area under a curve: Example Solved Problems with Answer, Solution

Example 3.1

Find the area bounded by *y* = 4*x* + 3 with *x*- axis between the lines *x* = 1 and *x* = 4

*Solution:*

Example 3.2

Find the area of the region bounded by the line *x *âˆ’* *2* y *âˆ’12* *=* *0* *, the* y*-axis and the lines* y *= 2,* y *= 5.

*Solution:*

*x *âˆ’* *2* y *âˆ’12* *=* *0

*x *=* *2* y *+* *12

Required Area

Example 3.3

Find the area of the region bounded by the parabola *y* = 4 âˆ’ *x*2 , *x* âˆ’ *axis* and the lines *x *=* *0,* x *=* *2

*Solutions:*

Example 3.4

Find the area bounded by *y* = *x* between the lines *x* = âˆ’1*and x* = 2 with *x* -axis.

*Solutions:*

Example 3.5

Find the area of the parabola *y* 2 = 8*x* bounded by its latus rectum.

*Solution*

y2=8x (1)

Comparing this with the standard form *y* 2 = 4*ax* ,

4*a* = 8

*a *=* *2

Equation of latus rectum is *x* = 2

Since equation (1) is symmetrical about *x*- axis

Required Area = 2[Area in the first quadrant between the limits *x* = 0 and *x* = 2]

Example 3.6

Sketch the graph *y* = |*x* + 3| and evaluate

*Solution:*

Example 3.7

Using integration find the area of the circle whose center is at the origin and the radius is a units.

*Solution*

Equation of the required circle is x2 + y2 = a2 (1)

put y = 0, x2 = a2

â‡’ x = Â± a

Since equation (1) is symmetrical about both the axes

The required area = 4 [Area in the first quadrant between the limit 0 and a.]

Example 3.8

Using integration find the area of the region bounded between the line *x* = 4 and the parabola *y* 2 = 16*x*.

*Solution:*

The equation *y* 2 = 16*x* represents a parabola (Open rightward)

Tags : Integral Calculus , 12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II

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12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II : The area of the region bounded by the curves: Example Solved Problems | Integral Calculus

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