Applications of the Integral

Problem : Find the area of the region in the plane between the graphs of f (x) = | x| and g(x) = - x ² + 6 .

Solution for Problem 1 >>

The intersection points of the graphs are at x = ±2 , and - x ² +6≥| x| on the interval [- 2, 2] , so the area of the relevant region is equal to

(- x 2 + 6 - | x|)dx	=	(- x 2 +6)dx - (- x)dx - xdx
	=	+ 6x - |-2 0 - |0 2
	=

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Problem : Find the total area between the graphs of f (x) = e ^x and g(x) = 1 + 2x from x = - 1 to x = 1 .

Solution for Problem 2 >>

Note that e ^x≥1 + 2x on the interval [- 1, 0] and 1 + 2x≥e ^x on the interval [0, 1] . Therefore the total area between the graphs is

| e x - (1 + 2x)| dx	=	(e x -1 - 2x)dx + (1 + 2x - e x)dx
	=	[e x - x - x 2]-1 0 + [x + x 2 - e x]0 1
	=	1 - + [(2 - e) - 1]
	=

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