To find the area under the curve Warm-Up: Graph. Area under a curve for [0, 3]  The area between the x-axis and the function Warm-up What is the area.

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Presentation transcript:

To find the area under the curve Warm-Up: Graph

Area under a curve for [0, 3]  The area between the x-axis and the function Warm-up What is the area under the function on the interval [0, 3]?

Find the area of the function from [0, 5]

Find the area under the function Problem???

Riemann Sums  Approximates the area under the curve using rectangles  Three different orientations for the rectangles left

Riemann Sums  Approximates the area under the curve using rectangles  Three different orientations for the rectangles left middle

Riemann Sums  Approximates the area under the curve using rectangles  Three different orientations for the rectangles left middle right

Riemann Sums  Sum the areas of all rectangle -the height of each rectangle is f(x) -the width of each rectangle is dx

Use Riemann sums to find the area under the curve

Left Riemann Sums Let dx = 0.5

Left Riemann Sum

Middle Riemann Sum

Right Riemann Sums

Right vs Left Riemann Sums

Determine the following  Given a function, an interval and dx Find: 1. Sketch the function and the approximating rectangles using the midpoints 2. Approximate the area under the curve

On the interval [0, 4] dx = 1 1. Sketch

2. Area