04Revisiting Kinematics and Areas Between Curves2

04Revisiting Kinematics and Areas Between Curves2 - curves...

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R EVISITING K INEMATICS AND A REAS B ETWEEN C URVES Kinematics Recall that the gradient (slope) of a displacement-time graph gives a velocity-time graph. Therefore, the area under a velocity-time graph gives a displacement-time graph and the area under an acceleration-time graph gives a velocity time graph. Using this process you can derive the formulas for velocity and displacement given uniformly accelerated motion (physics). Note: Displacement = signed area Distance = total area a b Ex . The velocity of an object is v m/s after t seconds, where t v 2 sin 2 1 - = . a) Find the object’s displacement over the first 4 3 π seconds. b) Find the object’s distance over the first 4 3 π seconds. (Absolute Value) (Determine t-intercepts)
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Areas Between Two Curves We can find the area between a curve and the x-axis, now we will find the area between two
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Unformatted text preview: curves. a b Ex . Find the area of the region bounded by the parabolas 2 x y = and 2 2 x x y-= . First find the intersection (endpoints). Look at the graph to determine which function is ) ( x f and which is ) ( x g . Ex . Find the area of the region enclosed by 2 y x = and 2-= x y . Draw a sketch because we have two separate regions. [ ] ∫ ∫ ∫-=-= b a b a b a dx x g dx x f A dx x g x f A ) ( ) ( ) ( ) ( 1 A 2 A (Note: 2 possible ways: Find endpoints in terms of x and integrate dx, or find endpoints in terms of y and integrate dy.) Assign: p. 417 # 5-17 odds, 27; Worksheet Kinematics #18-20; p.448 #5-17 odds, 31, 33...
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