34 Lecture

What is the formula for finding the volume of a disk using integration?

A. V = ?r^2

B. V = ?r^2h

C. V = ?h^2/4

D. V = ?h^2/2

Answer: B

When finding the volume of a washer using integration, what is the distance between the two radii?

A. The thickness of the washer

B. The diameter of the inner radius

C. The diameter of the outer radius

D. The difference between the two radii

Answer: A

What is the shape of the cross-section when finding the volume of a solid of revolution by slicing perpendicular to the axis of revolution?

A. Rectangle

B. Trapezoid

C. Triangle

D. Disk or washer

Answer: D

In which direction is the solid of revolution sliced when finding its volume using disks and washers?

A. Perpendicular to the axis of revolution

B. Parallel to the axis of revolution

C. Along the axis of revolution

D. Diagonal to the axis of revolution

Answer: A

What is the formula for finding the volume of a solid of revolution using disks?

A. V = ?r^2

B. V = ?r^2h

C. V = ?[a,b] f(x)^2 dx

D. V = ?[a,b] ?f(x)^2 dx

Answer: D

What is the formula for finding the volume of a solid of revolution using washers?

A. V = ?r^2

B. V = ?r^2h

C. V = ?[a,b] f(x)^2 dx

D. V = ?[a,b] ?(R^2-r^2) dx

Answer: D

When finding the volume of a solid of revolution using washers, what does the term "R" represent?

A. The radius of the solid at the outer edge of the washer

B. The radius of the solid at the inner edge of the washer

C. The radius of the washer itself

D. The thickness of the washer

Answer: A

When finding the volume of a solid of revolution using washers, what does the term "r" represent?

A. The radius of the solid at the outer edge of the washer

B. The radius of the solid at the inner edge of the washer

C. The radius of the washer itself

D. The thickness of the washer

Answer: B

What is the shape of the cross-section when finding the volume of a solid of revolution by slicing parallel to the axis of revolution?

A. Rectangle

B. Trapezoid

C. Triangle

D. Disk or washer

Answer: A

What is the formula for finding the volume of a solid of revolution by slicing parallel to the axis of revolution?

A. V = ?r^2

B. V = ?r^2h

C. V = ?[a,b] f(x) dx

D. V = ?[a,b] 2?xf(x) dx

Answer: D

What is the formula for finding the volume of a solid using the disk method?

Answer: The formula for finding the volume of a solid using the disk method is V = ??(a to b) [f(x)]^2 dx.

What is the formula for finding the volume of a solid using the washer method?

Answer: The formula for finding the volume of a solid using the washer method is V = ??(a to b) [R(x)^2 - r(x)^2] dx.

What is the difference between the disk method and the washer method?

Answer: The disk method is used when the cross-sections are disks, while the washer method is used when the cross-sections are washers.

What is the difference between the inner radius and the outer radius in the washer method?

Answer: The inner radius is the distance from the center of the cross-section to the inner edge, while the outer radius is the distance from the center of the cross-section to the outer edge.

How do you know when to use the disk method or the washer method?

Answer: You use the disk method when the cross-sections are disks, and the washer method when the cross-sections are washers.

What is the formula for finding the volume of a solid when the cross-sections are semicircles?

Answer: The formula for finding the volume of a solid when the cross-sections are semicircles is V = 1/2 ??(a to b) [f(x)]^2 dx.

What is the formula for finding the volume of a solid when the cross-sections are squares?

Answer: The formula for finding the volume of a solid when the cross-sections are squares is V = ?(a to b) [f(x)]^2 dx.

What is the difference between a horizontal slice and a vertical slice?

Answer: A horizontal slice is parallel to the x-axis, while a vertical slice is parallel to the y-axis.

How do you find the volume of a solid using the disk or washer method when the function is not given in terms of x?

Answer: You can use the formula V = ??(a to b) [f(y)]^2 dy for the disk method and V = ??(a to b) [R(y)^2 - r(y)^2] dy for the washer method.

Can the disk and washer method be used for any solid?

Answer: No, the disk and washer method can only be used for solids that can be sliced into disks or washers perpendicular to a given axis.

Volume by slicing; Disks and Washers

Volume by slicing is a technique used in calculus to find the volume of three-dimensional objects. In particular, disks and washers are two common shapes that are used to find the volume of a solid. The concept of slicing is based on the idea that if we divide a three-dimensional object into many thin slices, the volume of the object can be approximated by the sum of the volumes of each slice. By using integration, we can then find the exact volume of the object. Disks and washers are two shapes that can be used to find the volume of a solid that is formed by revolving a function about an axis. In order to use these shapes, we first need to determine the formula for the function that we want to revolve. Once we have this formula, we can use it to create a cross-sectional area of the object. This cross-sectional area is the area of a slice of the object at a given height. A disk is formed by revolving a function about an axis, where each cross-sectional area is a circle. The radius of the circle is equal to the value of the function at the given height. The formula for the volume of a solid formed by disks is: V = ??[a,b] (f(x))^2 dx

where f(x) is the function that is being revolved, a and b are the limits of integration, and dx represents the thickness of each slice.

A washer, on the other hand, is formed by revolving a function about an axis, where each cross-sectional area is a washer-shaped ring. The area of the ring is the difference between the area of the outer circle and the area of the inner circle. The radii of the outer and inner circles are equal to the value of the function at the given height plus and minus the thickness of the slice, respectively. The formula for the volume of a solid formed by washers is: V = ??[a,b] [(f(x)+g(x))^2 - (f(x))^2] dx where g(x) is the thickness of each slice, which can be a constant or another function of x. These formulas can be used to find the volume of a wide variety of solids. For example, the volume of a sphere can be found by revolving the function f(x) = ?(r^2 - x^2) about the x-axis, where r is the radius of the sphere. By using the formula for the volume of a solid formed by disks, we get: V = ??[-r,r] (f(x))^2 dx = ??[-r,r] (r^2 - x^2) dx = (4/3)?r^3 Similarly, the volume of a cone can be found by revolving the function f(x) = (r/h)x about the x-axis, where r is the radius of the base and h is the height of the cone. By using the formula for the volume of a solid formed by washers, we get: V = ??[0,h] [(r/h)x + r]^2 - (r)^2 dx = (1/3)?r^2h In conclusion, disks and washers are two shapes that are commonly used in calculus to find the volume of a solid formed by revolving a function about an axis. By using integration, we can find the exact volume of a wide variety of objects.