lecture32 - Math 121 Lecture 32 6.4 Areas in the xy-Plane !...

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§ 6.4 Areas in the xy -Plane Math 121 Lecture 32 ! Properties of Definite Integrals ! Area Between Two Curves ! Finding the Area Between Two Curves
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y = x 2 – 3 x and the x -axis ( y = 0) from x = 0 to x = 4. Upon sketching the graphs we can see that the two graphs cross; and by setting x 2 – 3 x = 0, we find that they cross when x = 0 and when x = 3. Thus one graph does not always lie above the other from x = 0 to x = 4, so that we cannot directly apply our rule for finding the area between two curves. However, the difficulty is easily surmounted if we break the region into two parts, namely the area from x = 0 to x = 3 and the area from x = 3 to x = 4. For from x = 0 to x = 3, y = 0 is on top; and from x = 3 to x = 4, y = x 2 – 3 x is on top. Consequently, area from x = 0 to x = 3 [ ] = 0 ( ) " x 2 " 3 x ( ) [ ] 0 3 # dx = " x 2 + 3 x ( ) 0 3 # dx = " x 3 3 + 3 x 2 2 # $ % ( 0 3 = " 9 + 27 2 # $ % ( " 0 " 0 ( ) = 4.5. Thus the total area is 4.5 + 1.833 = 6.333.
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lecture32 - Math 121 Lecture 32 6.4 Areas in the xy-Plane !...

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