To find the area under the curve

Warm-Up:Graph xxf 2

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

xf

Left Riemann Sums Let dx = 0.5

Left Riemann Sum

Middle Riemann Sum

Middle Riemann Sum

Right Riemann Sums

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 midpoints2. Approximate the area under the curve

On the interval [0, 4] 216 xxf dx = 1

1. Sketch

2. Area

216 xxf