AP Calculus Mrs. Mongold

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Presentation transcript:

AP Calculus Mrs. Mongold Lesson 2: Area Take 2 AP Calculus Mrs. Mongold

Note LRAM and RRAM are different then upper and lower sums!!!

When n=4 ,∆𝑥= 𝜋 4 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 8 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 3𝜋 4 +𝑠𝑖𝑛𝜋 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 8 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 3𝜋 4 +𝑠𝑖𝑛𝜋

When n=4 ,∆𝑥= 𝜋 4 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 8 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 3𝜋 4 +𝑠𝑖𝑛𝜋 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 8 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 3𝜋 4 +𝑠𝑖𝑛𝜋 ≈ 𝜋 8 0+ 2 +2+ 2 +0

When n=4 ,∆𝑥= 𝜋 4 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 8 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 3𝜋 4 +𝑠𝑖𝑛𝜋 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 8 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 3𝜋 4 +𝑠𝑖𝑛𝜋 ≈ 𝜋 8 0+ 2 +2+ 2 +0 ≈1.896

When n = 8, ∆𝑥= 𝜋 8 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 16 ( 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 8 +2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 3𝜋 8 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 5𝜋 8 +2𝑠𝑖𝑛 3𝜋 4 +2𝑠𝑖𝑛 7𝜋 8 +𝑠𝑖𝑛𝜋)

When n = 8, ∆𝑥= 𝜋 8 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 16 ( 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 8 +2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 3𝜋 8 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 5𝜋 8 +2𝑠𝑖𝑛 3𝜋 4 +2𝑠𝑖𝑛 7𝜋 8 +𝑠𝑖𝑛𝜋) ≈ 𝜋 16 2+2 2 +4𝑠𝑖𝑛 𝜋 8 +4𝑠𝑖𝑛 3𝜋 8

When n = 8, ∆𝑥= 𝜋 8 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥≈ 𝜋 16 ( 𝑠𝑖𝑛0+2𝑠𝑖𝑛 𝜋 8 +2𝑠𝑖𝑛 𝜋 4 +2𝑠𝑖𝑛 3𝜋 8 +2𝑠𝑖𝑛 𝜋 2 +2𝑠𝑖𝑛 5𝜋 8 +2𝑠𝑖𝑛 3𝜋 4 +2𝑠𝑖𝑛 7𝜋 8 +𝑠𝑖𝑛𝜋) ≈ 𝜋 16 2+2 2 +4𝑠𝑖𝑛 𝜋 8 +4𝑠𝑖𝑛 3𝜋 8 ≈1.974

Evaluate Using Antiderivative 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥

Evaluate Using Antiderivative 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥 - (𝑐𝑜𝑠𝜋−𝑐𝑜𝑠0)

Evaluate Using Antiderivative 0 𝜋 𝑠𝑖𝑛𝑥𝑑𝑥 - (𝑐𝑜𝑠𝜋−𝑐𝑜𝑠0) A=2

Verify your approximation with your calculator Graph f(x) 2nd Calc #7 - 𝑓 𝑥 𝑑𝑥 Enter upper and lower bounds