Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the
1 answer:
Answer:
Step-by-step explanation:
If you graph this on a Cartesian coordinate system, it resembles a logarithmic function. It begins at (0, 0) and concludes, based on the boundaries provided, at x = 8. Since we are finding the volume using the shell method, we need to establish x = y equations (which we fortunately have) as well as y intervals. The y-interval is specified in the problem as 0≤y≤2. When we rotate this solid around the line y = 2, since the axis of rotation is above the solid, we need to work backwards to reach there. This is crucial. The formula we apply for shells is
where p(y) refers to the distance from the axis of rotation to the solid, and h(y) signifies the height of the rectangle we represent. Notably, since we utilize the shell method, this rectangle is aligned parallel to the axis of rotation, thus, it has a height of 8 units.
Here, p(y) is the distance from y = 2 to the solid, hence p(y) = 2 - y
h(y) represents the rectangle's height, therefore h(y) = 8
and inserting values into the formula gives us:
and simplifying slightly:
Integrating yields
from 2 to 0. Leveraging the First Fundamental Theorem of Calculus results in:
which ultimately simplifies to
which is, finally,
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Answer:
Detailed explanation:
* Let’s clarify the procedure for solving this problem
- An exponential function can be expressed as , where
a signifies the starting amount (x = 0), and b denotes the growth factor
- If b > 1, it qualifies as an exponential growth function
- If 0 < b < 1, it is categorized as an exponential decay function
- Horizontal translation to the right by h units results in the new function being
- Conversely, if translated to the left by h units, the function modifies to
- A vertical upward shift by k units alters the function to
- A vertical downward shift by k units means the new function accounts for
* Now, let’s solve the problem
∵ f(x) is an exponential function
∵ The points (0, 1), (1, 10), (2, 100) are points of f(x)
- g(x) is the transformation of f(x)
∵ The points (3, 1), (4, 10), (5, 100) belong to g(x)
∵ The point (0, 1) on f(x) transforms to (3, 1) on g(x)
∵ The point (1, 10) on f(x) transforms to (4, 10) on g(x)
∵ The point (2, 100) on f(x) translates to (5, 100) on g(x)
∵ Notably, all y-coordinates of f(x) match those of g(x)
indicating no vertical translation
∴ Hence, no vertical translation occurred
∵ The x-coordinates of f(x) are increased by
3 units to yield the x-coordinates for g(x)
∴ This shows that f(x) is translated 3 units to the right
∵
∴
- Refer to the corresponding graph for greater clarity
# The red curve denotes f(x)
# The blue curve signifies g(x)
