Objective: Be able to approximate the area under a curve.

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Section 8.5 Riemann Sums and the Definite Integral.

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Presentation transcript:

Objective: Be able to approximate the area under a curve. Reimann Sums Objective: Be able to approximate the area under a curve. Warm-Up: What is the area of a trapezoid with bases 10 and 12 and a height of 5?

Area Under the Curve What is the area of the shaded region?

Area Under the Curve Approximate the area by dividing the region into rectangles.

Area Under the Curve To get a better approximation we could use more rectangles.

Area Under the Curve As the number of rectangles increases, the lengths of the individual bases approach zero. The more rectangles we put in, the closer we get to the actual area.

Area Under the Curve By taking a limit as the lengths of the bases approach zero, you combine an infinite number of rectangles. The sum of the area of all these rectangles equals the area of the actual region.

Left-Handed Approximation

Right-Handed Approximation

Midpoint Approximation

Approximating the area under the curve. Approximate the area under f(x) = 9 – x2 between the interval [-2, 2] using a right handed sum with n = 4.

Approximating the area under the curve. Approximate the area under f(x) = 3 – ¼ x3 Between the interval [-1, 2] using a left handed sum with n = 3. Is this an over or under estimate? What would a right handed sum be?

Approximating the area under the curve. Approximate the area under f(x) = x2 – x3 Between the interval [-1, 1] using a midpoint sum with n = 4.

Approximating the area under the curve. Approximate the area under f(x) = ¼ (x2 +4x) Between the interval [1, 4] using a trapezoidal sum n = 3. Is it an over or under estimate? What would a left handed sum be?