Areas & Riemann Sums TS: Making decisions after reflection and review.

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Presentation transcript:

Areas & Riemann Sums TS: Making decisions after reflection and review

Objective  To find the area of exotic shapes.

Two Questions of Calculus Q 1 : How do you find instantaneous velocity? A: Use the derivative. Q 2 : How do you find the area of exotic shapes? A: We don’t know yet.

Area Under the Curve Suppose we wanted to find the area between the curve and the floor.

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

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.

Riemann Sums  A Riemann sum is a method for approximating the total area underneath a curve on a graph.  We will examine three types of Riemann sums:  Left-handed approximation  Right-handed approximation  Midpoint approximation

Left-Handed Approximation

Right-Handed Approximation

Midpoint Approximation

Area Approximation Approximate the area between the curve f (x) = x and the x-axis on the interval [0, 2] using 4 rectangles.

Approximate the area between the curve f (x) = x and the x-axis on the interval [0, 2] using 4 rectangles. Using a Right Handed Approximation

Approximate the area between the curve f (x) = x and the x-axis on the interval [0, 2] using 4 rectangles. Using a Left Handed Approximation

Approximate the area between the curve f (x) = x and the x-axis on the interval [0, 2] using 4 rectangles. Using a Midpoint Approximation

Conclusion   When trying to find the area of a complicated region, try approximating the area with rectangles.   As the number of rectangles used to approximate the area of the region increases, the approximation becomes more accurate.