Approximation and Computing Area

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Presentation transcript:

Approximation and Computing Area AP Calculus 5-1 Approximation and Computing Area

𝑦= 𝑥 2 𝑜𝑛 0,4

Right-Endpoint Approximation With N intervals find the area under the curve 4 f(x) Area 16 64 2 f(x) Area 4 8 16 32 1 f(x) Area 2 4 3 9 16

¼ f(x) Area 1/16 1/64 ½ ¾ 9/16 9/64 1 5/4 25/16 25/64 3/2 9/4 7/4 49/16 49/64 2 4 81/16 81/64 5/2 25/4 11/4 121/16 121/64 3 9 13/4 169/16 169/64 7/2 49/4 15/4 225/16 225/64 16

Left-Endpoint Approximation With N intervals find the area under the curve 4 f(x) Area 2 f(x) Area 4 8 1 f(x) Area 2 4 3 9

Mid-Point Approximation With N intervals find the area under the curve 4 f(x) Area 2 16 2 f(x) Area 1 3 9 18 1 f(x) Area ½ ¼ 3/2 9/4 5/2 25/4 7/2 49/4

𝑓 𝑥 increasing 𝑓(𝑥) decreasing 𝐿 𝑁 ≤𝑎𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝑐𝑢𝑟𝑣𝑒≤ 𝑅 𝑁 𝑅 𝑁 ≤𝑎𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝑐𝑢𝑟𝑣𝑒≤ 𝐿 𝑁

𝐿 6 , 𝑓 𝑥 = 6𝑥+2 , 1,3 ∆𝑥= 3−1 6 = 1 3 7.1463 x f(x) Area 1 2.828 𝐿 6 , 𝑓 𝑥 = 6𝑥+2 , 1,3 ∆𝑥= 3−1 6 = 1 3 7.1463 x f(x) Area 1 2.828 0.9428 4/3 3.1622 1.0541 5/3 3.4641 1.1547 2 3.7417 1.2472 7/3 4 1.3333 8/3 4.2426 1.4142

Problems 5-1 Page 296 #1-12, 13-19 odd