Finding Volumes Of A Solid Flashcards
(21 cards)
When is disk method used?
When the defined region borders the axis of revolution over the entire interval (a, b)
Disk method revolving around x-axis
Volume = π ∫ [ f(x) ]² dx
Disk Method revolving around y-axis
Volume = π ∫ [ f(y) ]² dy
Disk Method revolving around horizontal line y = k
Volume = π ∫ [ f(x) - k ]² dx
Disk Method revolving around vertical line x = m
Volume = π ∫ [ f(y) - m ]² dy
How do we get volume if given Area f(x)
∫ f(x) dx
Integrate area from a to b
When is Washer Method used?
When the defined region has a space between the axis of revolution on the interval (a, b)
How many functions make up the Washer Method?
2 functions
When using washer method which should be first?
f(x)
The top function
Washer Method revolving around the x-axis
π ∫ [ f(x) ]² - [ g(x) ]² dx
Washer Method revolving around the y-axis
π ∫ [ f(y) ]² - [ g(y) ]² dy
Washer Method revolving around a horizontal line y = k
π ∫ [ f(x) - k ]² - [ g(x) - k ]² dx
Washer Method revolving around a vertical line x = m
π ∫ [ f(y) - m ]² - [ g(y) - m ]² dy
When is Cross Sections used
When a defined region is used as the base of a solid
When are cross sections in terms of X?
For cross sections perpendicular to the X-axis and a region bounded by f and the axis
Cross Sections are Squares
V = ∫ [ f(x) ]² dx
Cross Sections are Equilateral Triangles
V = √3/4 ∫ [ f(x) ]² dx
Cross Sections are Isosceles Right Triangles with a Leg in the base
V = 1/2 ∫ [ f(x) ]² dx
Cross Sections are Isosceles Right Triangles with a Hypotenuse in the base
V = 1/4 ∫ [ f(x) ]² dx
Cross Sections are Semi Circles with Diameter in the base
V = π/8 ∫ [ f(x) ]² dx
Cross Sections are Semi Circles with Radius in the base
V = π/2 ∫ [ f(x) ]² dx
