Application of Integrals
Madhya Pradesh Board · Class 12 · Mathematics
Summary of Application of Integrals for Madhya Pradesh Board Class 12 Mathematics. Key concepts, important points, and chapter overview.
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Application of Integrals connects definite integrals with area measurement. The main idea is that area under a curve can be found by adding the areas of many thin strips. For a curve y = f(x), the elementary area of a thin vertical strip is dA = y dx, so the total area is A = \int_a^b dA = \int_a^b
Key Concepts
A thin vertical strip of height
A thin vertical strip of height y and width dx has elementary area dA = y dx. This is the basic idea used to build area by integration.
The area bounded by the curve
The area bounded by the curve y = f(x), the x-axis, and the ordinates x = a and x = b is A = \int_a^b f(x) dx.
The area bounded by the curve
The area bounded by the curve x = g(y), the y-axis, and the lines y = c and y = d is A = \int_c^d g(y) dy.
If a curve lies below
If a curve lies below the x-axis over an interval, direct integration gives a negative value. The actual area is the absolute value, \left| \int_a^b f
If part of a curve lies
If part of a curve lies above the x-axis and part lies below it, the region must be split. The area is A = |A_1| + A_2.
Learning Objectives
- Find area under a curve using vertical strips.
- Find area under a curve using horizontal strips.
- Handle regions below the x-axis correctly by using absolute value.
- Find area of regions partly above and partly below the x-axis by splitting the interval.
- Calculate area enclosed by a circle and an ellipse using symmetry.
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