Week 16 - Applications of Integration
1. Area between Curves
- The area between the curves and and between and is
- For example,
- we can simply get the integral function
- Some regions are best treated by regarding x as a function of y. If a region is bounded by curves with equations , , , and , where f and g are continuous and for , then its area is
- For example,
- If treat them as functions of x, we need to split the area to two parts.
- So we rewrite the functions to
2.1. Sphere's Volume
- The cross-sectional area is
- Using the definition of volume with and , we have
2.2. Use Washer Method
- The region enclosed by the curves is rotated about the x-axis. Find the volume of the resulting solid.
- The curves intersect at the points (0, 0) and (1, 1). The region between them, the solid of rotation, and a cross-section perpendicular to the x-axis are shown above. A cross-section in the plane has the shape of a washer (an annular ring) with inner radius and outer radius , so we find the cross-sectional area by subtracting the area of the inner circle from the area of the outer circle:
- Therefore we have
2.3. Use Shells Method
- To face a situation like below. If we slice perpendicular to the y-axis, we get a washer. But to compute the inner radius and the outer radius of the washer, we’d have to solve the cubic equation for x in terms of y; that’s not easy.
- For this situation, we use the method of cylindrical shells.
- Instead of slicing the perpendicular to the x-axis:
- Then, flatten as below:
- Radius , circumference , height , and thickness or :
- For example: Find the volume of the solid obtained by rotating the region bounded by and about the line .
- The figure below shows the region and a cylindrical shell formed by rotation about the line . It has radius , circumference , and height .
- The volume of the given solid is:
2.4. Disks and Washers versus Cylindrical Shells
- If the region more easily described by top and bottom boundary curves of the form , or by left and right boundaries , use Washers.
- If we decide that one variable is easier to work with than the other, then this dictates which method to use.
- Draw a sample rectangle in the region, corresponding to a cross-section of the solid. The thickness of the rectangle, either or , corresponds to the integration variable. If you imagine the rectangle revolving, it becomes either a disk (washer) or a shell.
3. Arc Length
- Formula: If is continuous on [a, b], then the length of the curve , is
- Suppose that a curve C is defined by the equation , where is continuous and . We obtain a polygonal approximation to C by dividing the interval [a, b] into n subintervals with endpoints and equal width . If , then the point lies on C and the polygon with vertices , is an approximation of C.
- We define the length L of the curve C as thee limit of the lengths :
- If we let , then