16bf10lec5 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b Areas as integrals If f ( x ) is a function of one variable for which f ( x ) ≥ 0 for all numbers x in the range a ≤ x ≤ b , then the area of the region bounded by the lines y = 0, x = a and x = b and the curve y = f ( x ) is given by the integral R b a f ( x ) dx Thomas Scanlon Math 16b One variable example Compute the area bounded the lines x = 1, x = 3, y = 0 and y = x 3- 3 x 2 + 5. Figure: y = x 3- 3 x 2 + 5 Thomas Scanlon Math 16b Solution Z 3 1 ( x 3- 3 x 2 + 5 ) dx = Z 3 1 x 3 dx- 3 Z 3 1 x 2 dx + 5 Z 3 1 dx = ( 1 4 x 4- x 3 + 5 x ) | x = 3 x = 1 = 81 4- 27 + 15- 1 4 + 1- 5 = 4 Thomas Scanlon Math 16b Volume integrals More generally, if F ( x , y ) is a function of two variables and over some region R in the plane, F is never negative, then the volume of region bounded by the graph of y = F ( x , y ) , the x- y coordinate plane, and boundary of the region is the double integral Z Z R F ( x , y ) dxdy Thomas Scanlon Math 16b Example...
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