AB Calculus
AP Calculus BC
7-1 Area of Region Between 2 Curves
Now, we will extend the application of definite integrals from area of a region under a curve to the area of a region between 2 curves.
Example: Consider 2 functions f and g that are continuous on the interval [a, b]
1. Set up the integral that represents the area of the region bounded by the graphs of [pic]
2. Set up the integral that represents the area of the region bounded by the graphs of [pic] and [pic]
3. Use your calculator: Find the area between the curves [pic]and [pic].
Area in terms of y…horizontal rectangles!
4. Find the area of the region bounded by the graphs of [pic] and [pic]
[pic]
HW: p.452-453 #1-6all, (17-23 odd, 27, 29 by hand), (33, 35, 37 graph & evaluate on calc), (49, 51 graph on calc, evaluate by hand)
AP Calculus BC
7-2 Volume
Key Terms:
If a region in the plane is revolved about a line, the resulting solid is a solid of revolution and the line that it is being revolved around is called the axis of revolution.
Simplest solid – a right circular cylinder or disk which is formed by revolving a rectangle about an axis adjacent to one side of the disc.
1. Find the volume of the solid formed by revolving the region bounded by the graph of [pic], the x-axis, and x=2 about the x-axis.
2. Find (set up) the volume of the solid formed by revolving the region bounded by [pic] and [pic] about the line [pic].
3. Find (set up) the volume formed by revolving the region bounded by the graph of [pic] and the y-axis about the y-axis.
7-2 Volume: The Washer Method
Washer - formed by revolving a rectangle about an axis that has both an inner and an outer radius
Note: The integral involving the inner radius represents the volume of the hole and is subtracted from the integral involving the outer radius.
1. Find (set up) the volume formed by revolving the region bounded by the graphs of [pic] and [pic] about the x-axis.
2. Find (set up) the volume formed by revolving the region bounded by the graphs of [pic] and [pic] about the x-axis.
3. Find (set up) the volume formed by revolving the region bounded by the graphs of [pic] about the y-axis.
Day 1(disk): p.463-464 #1-4all (just set up), 7, 9, 23-27odd, 31, (33&35 on calculator)
Day 2 (Washer): p.463-464 #(5, 6) just set up, 11ab, 13a, 29, 37 *For all: use calculator at your discretion!
4. Set up the integral that would find the volume formed by revolving the region bounded by the graphs of [pic] about:
a) y-axis b) x-axis
c) x=2 d) x=3
e) y=9 f) y= - 2
Day 3 HW: p.463-464 #15-21 odd
AP Calculus BC
7-3 – Volume: The Shell Method
Shell Method
• Helpful when revolving around y-axis or other vertical line
• Eliminates the need to change equation in terms of y
• Can replace disk or washer method, but isn’t always convenient
• Rectangles/shells are parallel to axis of revolution
• Not required on AP exam, but is acceptable and may be easier!
1. Derive the set-up for shells using the region bounded by [pic]revolved about the y-axis.
2. [pic]about y-axis 3. [pic]about x=3
4. [pic] about x=-1
[pic]
AP Calculus BC
7-2 Volume: Solids with Known Cross Sections
Sometimes, you will be given an object where you know the shape of the base and where perpendicular cross-sections are all the same, regular, planar geometric shape.
1. Find the volume of a solid whose base the region bounded by the graphs of [pic],[pic], and x=0 and where cross sections perpendicular to the x-axis are all squares whose sides lie on the base of the region.
[pic]
[pic]
Find the volume of a solid whose base the region bounded by the graphs of [pic] and [pic]and where cross sections perpendicular to the x-axis are all equilateral triangles whose sides lie on the base of the region.
HW: p. 465 #61 and first two FR
AP Calculus BC
7-4 Arc Length & Surface Area
Arc Length: As long as f is smooth (differentiable) on the interval [a, b], the Pythagorean Theorem shows us the length of each secant line…
1. Evaluate on the calculator.
2. Find the arc length of the curve on the given interval. Evaluate by hand.
3. Set up the integral to find the arc length of the function on the given interval.
4. Find the arc length. Evaluate by hand.
Surface Area
1. Set up and evaluate the definite integral for the area of the surface generated by revolving the given curve about the x-axis.
2. Set up the definite integral for the area of the surface generated by revolving the given curve about the
y-axis.
HW: p. 483-484 Arc Length: #3, 5, 9 by hand, 7 on calc, 11-19 odd on calc,
Surface Area: #39 by hand, 43 in terms of x, calc, 55 calc
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Area of a region Between 2 Curves
If f and g are continuous on [a, b] and [pic] for all x in [a, b], then the area of the region bounded by the graphs of f and g and vertical lines x = a and x = b is
[pic]
The Disk Method
To find the volume of a solid of revolution with the disk method, use the following
Horizontal axis of Revolution Vertical axis of revolution
[pic] [pic]
Formula for Washers [pic]
Steps:
1. Find the side of the cross section in terms of y. (This will involve a vertical slice)
2. Plug the side into the equation for area of the cross-section
3. Integrate the area from one endpoint of the base to the other.
[pic]
Length of Curve
[pic]
[pic]
[pic]
[pic]
Surface Area about x-axis:
R
[pic]
[pic]
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