In the formula V = integral (a, b) A(u)du used to compute the volume of a solid:
A. the du can be thought of as representing the thickness of the axis of revolution
B. the function A(u) represents cross-sectional area
C. the function A(u) represents the area of the intersection between a plane (perpendicular to an axis) and the solid.
D. A and B
E. B and C
not A - no mention was made of revolution
So I'd say E
E. B and C
Clown Bot: Hey there! Looks like we have a multiple-choice question here. Now, let's break it down, shall we?
We are talking about the formula V = integral (a, b) A(u)du, which is used to compute the volume of a solid. Now, the question is asking us about the different parts of this formula.
Option A says that the "du" represents the thickness of the axis of revolution. Well, unfortunately, "du" is not about the thickness of any axis. It actually represents an infinitesimal change in the variable of integration. So, this option is out.
Option B states that the function A(u) represents the cross-sectional area. Well, this option is definitely true! The function A(u) gives us the area of each cross-section when integrated.
Option C says that the function A(u) represents the area of the intersection between a plane (perpendicular to an axis) and the solid. Right on the money! When we integrate A(u), we're basically adding up all those cross-sectional areas, which indeed represents the intersection between the plane and the solid.
So, we've found that Option B and C are both correct. Hence, the answer is E. B and C.
Did that make you laugh and learn at the same time? I sure hope so!
E. B and C
In the formula V = integral (a, b) A(u)du, the function A(u) represents the cross-sectional area of the solid at a given position along the axis of revolution. This means that as the value of u varies from a to b, the function A(u) provides the varying cross-sectional area of the solid.
Additionally, the function A(u) can be understood as representing the area of the intersection between a plane perpendicular to an axis and the solid. By integrating this function over the interval (a, b), we sum up these cross-sectional areas to compute the volume of the solid.
E. B and C
To understand why, let's break down the components of the formula:
V = integral (a, b) A(u)du
- The integral symbol (∫) indicates that we are calculating the definite integral of a function.
- The limits of integration, a and b, represent the boundaries within which we want to calculate the volume.
- The function A(u) represents the cross-sectional area of the solid at a given position u along the axis of revolution.
The key points to note are:
B. The function A(u) represents the cross-sectional area of the solid. This means that for each value of u, A(u) tells us the area of the cross-section of the solid at that position. This concept is crucial in determining the volume because we are essentially summing up all the infinitesimally small cross-sectional areas along the axis of revolution.
C. The function A(u) represents the area of the intersection between a plane (perpendicular to an axis) and the solid. As we move along the axis of revolution, each cross-section of the solid intersects with a plane perpendicular to the axis. The area of this intersection, given by A(u), is what we need to calculate in order to find the volume.
So, both statements B and C are correct. The function A(u) represents both the cross-sectional area of the solid and the area of the intersection between a plane and the solid. Therefore, the answer is E. B and C.