5.1 Calculus and Area.

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Presentation transcript:

5.1 Calculus and Area

I. Problem! Given a nonnegative function f (x), i.e., f (x) > 0, on [a, b]. Find the area bounded by the curve, the x- axis, and the vertical lines x = a and x = b.

II. Estimation of the Area A.) Partition [a, b] into n subintervals of equal length by choosing x-values

B.) Evaluate the area for given n using rectangles: Rectangular Approximation Method (RAM) Left-hand approximation:

Right-hand approximation: Mid-point approximation:

C.) Visually: 1 2 3 4 LRAM

1 2 3 4 RRAM

1 2 3 4 MRAM

III. Solution – Area Under a Curve Def.- If f (x) ≥ 0 and is continuous on [a, b], then the area bounded by the curve, the x-axis, and the vertical lines x = a and x = b is

IV. Examples of Approximations of Area A.) Approximate the area under the following curve from x = 1 to x = 3 using 4 rectangles and all three RAMs.

LRAM

RRAM

MRAM

B.) Approximate the area under the following curve from x = 0 to x = 2 using 10 rectangles and all three RAMs.

Is LRAM always an underestimate(lowest)? C.) Approximate the area under the following curve from x = 0 to x = 5 using 3 rectangles and all three RAMs.