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Applications of Integration | Mathematics - Summary | 12th Maths : UNIT 9 : Applications of Integration

Chapter: 12th Maths : UNIT 9 : Applications of Integration

Summary

Maths : Applications of Integration: Summary

SUMMARY

(1) Definite integral as the limit of a sum


(2) Properties of definite integrals


(3) Bernoulli’s Formula


(4) Reduction Formulae


(5) Gamma Formulae


(6) Area of the region bounded by a curve and lines

(i) The area of the region bounded by a curve, above x -axis and the lines x = a and x = b is A = ba y dx .

(ii) The area of the region bounded by a curve, below x -axis and the lines x = a and x = b is A = −ba ydx = | ba ydx | .

(iii) Thus area of the region bounded by the curve to the right of y -axis, the lines y = c and y = d is A = dc  xdy .

(iv) The area of the region bounded by the curve to the left of y -axis, the lines y = c and y = d is A = − dc xdy = | dc xdy |.

(v) If f (x) ≥ g(x), then area bounded by the curves and the lines x = a, x = b is

A = ba [ f (x) − g( x)]dx = ba ( yU − yL ) dx

(vi) If f (y) ≥ g(y), then area bounded by the curves and the lines y = c, y = d is

A = dc [ f ( y) − g(y)]dy = dc ( xR − xL ) dy

(7) Volume of the solid of revolution

(i) The volume of the solid of revolution about x-axis is V = π ba y2 dx.

(ii) The volume of the solid of revolution about y-axis is V = π dc x2 dy.


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12th Maths : UNIT 9 : Applications of Integration


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