Problem : Find the area of the region in the plane between the graphs of f (x) = | x| and g(x) = - x ^{2} + 6 .
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} | ||
= |
Problem : Find the total area between the graphs of f (x) = e ^{x} and g(x) = 1 + 2x from x = - 1 to x = 1 .
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] | ||
= |