Download Applications of the Definite Integral

Review Problem – Riemann Sums
 Use a right Riemann Sum with 3
subintervals to approximate the
definite integral:
3
 x dx
2
0
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 integration
2. Identify “top” curve & “bottom” curve OR
“right-most” curve & “left-most” curve
3. Draw a representative rectangle
4. 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 revolution
2. 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 xaxis is a square. Find the volume of the solid.
Homework #3 – Different axis &
known base
 P.249: 15,16,18,19