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Review Problem – Riemann Sums
Use a right Riemann Sum with 3 subintervals to approximate the definite integral:
x 2dx0
3
Applications of the Definite Integral
Mr. Reed
AP Calculus AB
Finding Areas Bounded by Curves
To get the physical area bounded by 2 curves:1. Graph curves & find intersection
points – limits of integration2. Identify “top” curve & “bottom”
curve OR “right-most” curve & “left-most” curve
3. Draw a representative rectangle4. Set up integrand: Top – Bottom Right – Left
Finding Intersection Points
Set equations equal to each other and solve algebraically
Graph both equations and numerically find intersection points
Example #1
Find the area of the region between y = sec2x and y = sinx from x = 0 to x = pi/4.
Example #2
Find the area that is bounded between the horizontal line y = 1 and the curve y = cos2x between x = 0 and x = pi.
Example #3
From Text – p.240 - #16
Example #4
Find the area of the region R in the first quadrant that is bounded above by y = sqrt(x) and below by the x-axis and the line y = x – 2.
Summarize the process
AP MC Area Problem
#12 from College Board Course Description
Homework
P.236-240: Q1-Q10, 13-25(odd)
Authentic Applications for the Definite Integral
Example #2 – p.237
Definite Integral Applied to Volume
2 general types of problems:1.Volume by revolution2.Volumes by base
Volume by Revolution – Disk Method
The region under the graph of y = sqrt(x) from x = 0 to x = 2 is rotated about the x-axis to form a solid. Find its volume.
Volume by Revolution – Disk Method
Homework #1 – Disk Method about x and y axis
P.246-247: Q1-Q10,1,3,5
Volume by Revolution – About another axis
The region bounded by y = 2 – x^2 and y = 1 is rotated about the line y = 1. Find the volume of the resulting solid.
Volume by Revolution – Washer Method
Find the volume of the solid formed by revolving the region bounded by the graphs of f(x) = sqrt(x) and g(x) = 0.5x about the x-axis.
Homework #2 – Washer Method & Different axis
P.247 – 249: 7,9,11,14
Volume with known base
The base of a solid is given by x^2 + y^2 = 4. Each slice of the solid perpendicular to the x-axis is a square. Find the volume of the solid.
Homework #3 – Different axis & known base
P.249: 15,16,18,19
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