What shape are the cross-sections in the washer method?
Washers
What does \( R(x) \) represent in the washer method formula for revolving around the x-axis?
Outer radius
The disk method assumes that the cross-sections are solid circles
What additional factor does the washer method account for that the disk method does not?
Inner radius
If the region between \( y = x \) and \( y = x^2 \) is revolved around the x-axis, the outer radius is x
The washer method calculates the volume of a solid formed by revolving a region between two curves around an axis
When revolving around the y-axis, the volume is calculated using the variable y
The washer method is used when cross-sections of a solid of revolution have a hole
Steps to set up the washer method integral when revolving around the x-axis:
1️⃣ Identify the outer and inner radii
2️⃣ Determine the limits of integration
3️⃣ Substitute into the formula
When revolving around the y-axis, the outer radius is denoted as R(y).
True
The antiderivative of x^2 is x^3/3.
True
Match the formula with the axis of revolution:
X-axis ↔️ V=π∫ab[R(x)2−r(x)2]dx
Y-axis ↔️ V=π∫cd[R(y)2−r(y)2]dy
When revolving around the x-axis, the volume formula includes the outer radius R(x)
How does the washer method differ from the disk method in terms of cross-sections?
Washer accounts for hole
The volume formula for revolving around the y-axis includes the term \( dy \)
True
When should the washer method be used instead of the disk method?
Cross-sections have a hole
Revolving the region between \( y = x \) and \( y = x^2 \) around the x-axis requires the washer method
True
What must be expressed in terms of \( y \) when revolving around the y-axis?
x
The example calculation results in a volume of 2π/15
True
The antiderivative of \( x^2 - x^4 \) is \(\frac{x^3}{3} - \frac{x^5}{5} + C\)
When revolving around the x-axis, the volume formula involves integrating \( [R(x)^2 - r(x)^2] \) with respect to x
When revolving around the y-axis, the radii \( R(y) \) and \( r(y) \) must be expressed as functions of \( y \).
True
When revolving around the y-axis, if \( x = \sqrt{y} \) is the outer radius and \( x = 0 \) is the inner radius, the limits of integration are \( y = 0 \) and y = 4
The volume of a solid of revolution around the x-axis using the washer method is given by π∫ab[R(x)2−r(x)2]dx
What axis of revolution is used when the volume formula includes \( dy \)?
y-axis
The washer method is used when the cross-sections of the solid have a hole in the center.
True
Steps to set up the integral for the washer method when revolving around the x-axis
1️⃣ Identify the outer and inner radii, \( R(x) \) and \( r(x) \)
2️⃣ Determine the limits of integration, \( a \) and \( b \)
3️⃣ Substitute into the volume formula
When revolving around the x-axis using the washer method, the variable of integration is \( x \).
True
In the washer method, R(x) represents the outer radius.
True
Match the method with its cross-sectional shape:
Disk ↔️ Circles
Washer ↔️ Washers
The disk method is used when cross-sections are solid circles.
True
The limits of integration in the washer method are determined by finding the points of intersection
To find the volume using the washer method, the first step is to simplify the integrand
The washer method differs from the disk method because it accounts for a hole in the cross-section
What shape does the washer method assume for the cross-sections of the solid being revolved?
Washer shape
When revolving around the y-axis, the volume formula uses \( R(y) \) as the outer radius
True
When revolving around the x-axis, the inner radius is represented by r(x)
Match the method with its cross-sectional shape:
Disk ↔️ Circle
Washer ↔️ Washer
The disk method uses the formula \( V = \pi \int [f(x)]^2 dx \) for solid shapes
Steps to set up the washer method integral when revolving around the x-axis: