Lecture19| 1
Chapter 7
Applications of Integration
7.1. Area of Region between two curves 7.2. Volume: the disk method
7.3. Volume: the shell method 7.4. Arclength of curve
Lecture19| 2 7.1. Area of Region between two curves
Consider the graph of . The region
under the curve from to is a
box of width and height , so the area
The integral
so
We had discussed the area problem before, and the conclusion is the following formula:
Lecture19| 3 Area under the curve. If is a continuous function from to , the area under the curve from to is
Here the region is bounded by
Lecture19| 4 This formula can be generalized to
Area of region between two curves.
If from to , then the area
of the region between the curves and from to is
Alternatively, set ,
Lecture19| 5
Parabola.
Standard. (I)
and parallel to -axis.
Lecture19| 6 (II)
and parallel to -axis.
Nonstandard.
Lecture19| 7
Lecture19| 8
EXAMPLE. Find the area of the region bounded
by the graphs of , and
.
Lecture19| 9
EXAMPLE. Find the area of the region bounded
by the graphs of , ,
and .
Lecture19| 10 Most practical problems, the two curves
and are intersecting, and we would like to get the area of the region bounded by the intersections. So, we must solve the equation
to get and at the intersection points.
It is also possible that the top function or the bottom function can change, so split the region!
Lecture19| 11 EXAMPLE. Find the area of the region bounded
by the graph and .
Lecture19| 12 EXAMPLE ( ). Find the area of the region
bounded by , -axis, and .
Lecture19| 13 For a region that has
where for all from to , the area is given by the formula
Alternatively, set and ,
Sometimes we must find the intersections of and to find and , or, we may need to split the region.
Lecture19| 14 EXAMPLE. Find the area of the region bounded
by and .
Lecture19| 15 EXAMPLE. Find the area of the region enclosed
by , , and -axis.
