Applications of Integration | Mathematics - Summary | 12th Maths : UNIT 9 : Applications of Integration
Chapter: 12th Maths : UNIT 9 : 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 = b∫a
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 = −b∫a ydx
= | b∫a 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 = d∫c
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 = − d∫c xdy = | d∫c
xdy |.
(v) If f (x) ≥ g(x), then area bounded by the
curves and the lines x = a,
x = b is
A = b∫a
[ f (x) − g( x)]dx = b∫a ( yU − yL ) dx
(vi) If f (y)
≥ g(y), then area bounded by the curves and the lines y = c, y = d
is
A = d∫c
[ f ( y) − g(y)]dy
= d∫c ( xR − xL ) dy
(7) Volume of the solid of revolution
(i) The
volume of the solid of revolution about x-axis
is V = π b∫a y2 dx.
(ii) The
volume of the solid of revolution about y-axis
is V = π
d∫c x2 dy.
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