week12 - Area between curves Definition : Let y 1 = f 1 ( x...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Area between curves Definition : Let y 1 = f 1 ( x ) and y 2 = f 2 ( x ) , x [ a, b ] , be two curves such that f 1 ( x ) > f 2 ( x ) for all x [ a , b ] . Let us then define y upper = f 1 ( x ) and y lower = f 2 ( x ). The area of the region bounded by the curves f 1 ( x ) and f 2 ( x ) and the vertical lines x = a and x = b is given by: Area = b Z a ( y upper- y lower ) dx = b Z a ( f 1 ( x )- f 2 ( x )) dx, (1) where y upper = y upper ( x ) is the curve higher (graphically) and y lower = y lower ( x ) is the curve lower (graphically), for all x [ a, b ]. Note 1 : y upper ( x ) and y lower ( x ) do not intersect for all x ( a, b ). Note 2 : The area is always a positive number. Special Cases : y upper = x-axis or y lower = x-axis. The equation of the x-axis is y = y ( x ) = 0. There are 2 cases to consider. Case 1 : Let y = f ( x ) be a curve such that all y-values are positive for all x [ a, b ], i.e. f ( x ) 0 for all x [ a , b ]. Then y upper = f ( x ) and y lower = x-axis, i.e. y lower = 0. The area between the curve y = f ( x ), the x-axis, the vertical lines x = a and x = b is thus: Area (above the x-axis) = b Z a ( y upper- y lower ) dx = b Z a [ f ( x )- 0] dx, by (1) i.e. Area (above the x-axis) = b Z a f ( x ) dx. (2) Example (Ex 6, Sect 4.7) Evaluate the area of the region bounded by y = 3 x 2 , the x-axis and the lines x = 1 and x = 3 . Solution All the y s on the curve y = 3 x 2 are positive for all x [1 , 3] . Since equation of x-axis is y ( x ) = y = 0 and thus y lower = 0, the area of the given region is simply: Area = 3...
View Full Document

Page1 / 10

week12 - Area between curves Definition : Let y 1 = f 1 ( x...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online