4.2 day 2 Riemann Sums Day 1.notebook
Jan 21­8:03 AM
January 28, 2015
Jan 21­8:17 AM
Area under a curve
• The area under a curve can be approximated by finding the area of
rectangles that lie under or over the curve
4.2 Area under a curve
Jan 20­12:52 PM
Jan 20­12:52 PM
Estimate the area
Rectangles
• The more rectangles we use, the closer our estimation will be to the
actual area.
Jan 20­12:52 PM
Jan 20­12:52 PM
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4.2 day 2 Riemann Sums Day 1.notebook
January 28, 2015
Estimate the area
Estimate the area
Jan 20­12:52 PM
Estimate the area
Jan 20­12:52 PM
Right Rectangles­Right Sum
• The height of the rectangle is on the right side
• This will underestimate the area in this case
Jan 20­12:52 PM
Jan 20­12:52 PM
Notice the following:
Jan 20­12:52 PM
Left Rectangle­Left Sum
• The height of the rectangle is on the left side
• This will overestimate the area in this case
Jan 20­12:52 PM
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4.2 day 2 Riemann Sums Day 1.notebook
January 28, 2015
Right and Left Rectangles
These are called Riemann Sums
Jan 20­12:52 PM
Jan 20­12:52 PM
Right Sum
Left Sum
Jan 20­12:52 PM
Jan 20­12:52 PM
Ex. 2 Approximate the left and right sum for the
shaded region(use 6 rectangles)
• f(x)=x2+3 on [2,5]
Jan 20­12:52 PM
Jan 20­12:52 PM
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4.2 day 2 Riemann Sums Day 1.notebook
Midpoint Rectangles
January 28, 2015
Ex. 2
• In the case of midpoint rectangles, you have to find the midpoint
between x0,x1,x2,…,xn. The midpoint between any 2 x values is
their sum divided by 2, so you will use:
• Find the midpoint sums for
rectangles.
Jan 20­12:52 PM
on the interval [2,6] with 4
Jan 20­12:52 PM
Trapezoids
Jan 20­12:52 PM
Jan 20­12:52 PM
Ex. 3
• Find the trapezoidal sum for
with 4 trapezoids.
on the interval [2,6]
Jan 20­12:52 PM
Jan 26­3:46 PM
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4.2 day 2 Riemann Sums Day 1.notebook
January 28, 2015
Ex 4.
• A car comes to a stop 5 seconds after the driver slams on the
brakes. While the brakes are on, the following velocities are
recorded. Estimate the total distance the car took to stop
(trapezoids).
Jan 22­8:32 AM
Jan 5­1:10 PM
Ex. 5
• You jump out of an airplane. Before your parachute
opens, you fall faster and faster. You acceleration
decreases as you fall due to air resistance. The table
below gives your acceleration a (in m/sec2) after t
seconds. Estimate the velocity after 5 seconds.
Jan 5­1:10 PM
Jan 26­3:44 PM
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