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Chapter 5 | Integration
Find the net signed area between the curve of the function f ( x ) =2 x and the x -axis over the interval [−3, 3].
Solution The function produces a straight line that forms two triangles: one from x =−3 to x =0 and the other from x =0 to x =3 ( Figure 5.19 ). Using the geometric formula for the area of a triangle, A = 1 2 bh , the area of triangle A 1 , above the axis, is A 1 = 1 2 3 (6) =9, where 3 is the base and 2(3) =6 is the height. The area of triangle A 2 , below the axis, is A 2 = 1 2 (3)(6) =9, where 3 is the base and 6 is the height. Thus, the net area is ∫ −3 3 2 xdx = A 1 − A 2 =9−9=0.
Figure 5.19 The area above the curve and below the x -axis equals the area below the curve and above the x -axis.
Analysis If A 1 is the area above the x -axis and A 2 is the area below the x -axis, then the net area is A 1 − A 2 . Since the areas of the two triangles are equal, the net area is zero.
Find the net signed area of f ( x ) = x −2 over the interval ⎡ ⎣ 0, 6 ⎤
⎦ , illustrated in the following image.
5.9
Total Area One application of the definite integral is finding displacement when given a velocity function. If v ( t ) represents the
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