Integ calc 2
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57 Terms
Surface Area [Pappus-guldin Theorems]
The resulting surface area of revolutions is equal to the product of the length of the curve and the displacement of its centroid
General formula in solving for the surface area
S=length arc
R=distance from centroid to axis of rotation
Derived formula for solving the surface area
arc length of a curve
R, distance from centroid to about x-axis
R=y
R, distance from centroid to about y-axis
R=x
R, distance from centroid to about y = ± value
R = y ∓ value
R, distance from centroid to about x = ±value
R = x ∓ value
Volume [Pappus-Guldin Theorems]
The resulting volume of revolution is equal to the product of the plane area enclosed by the curve and the displacement of the centroid of this area
General formula in solving for the volume
Derived formula for solving the volume
Shell Method
Axis of Rotation (x-axis) parallel to the strip
Washer Method/Disk Method
Axis of Rotation (x-axis) perpendicular to the strip
R in AOR parallel to the strip about the x-axis (horizontal strip)
R in AOR perpendicular to the strip about the x-axis (vertical strip)
R in AOR perpendicular to the strip about the y-axis (vertical strip)
R = XR-XL/2 = 2X/2 = x
Shell Method
axis of rotation parallel to the strip
Washer/Disk Method
axis of rotation perpendicular to rotation
R in AOR perpendicular to the strip about the y-axis (horizontal strip)
R = XR-XL/2
Moment
Measure the tendency of the region to rotate about the x and y-axis, respectively
Formula for Moment
Original Formula for Moment of Area
Original Formula for Moment of Volume
Formula for density of area
Formula for density of volume
General formula for Moment of area about an axis
General formula for Moment of volume about an axis
Moment about x-axis (vertical strip)
Moment about y-axis (vertical strip)
Moment about x-axis (horizontal strip)
Moment about y-axis (horizontal strip)
Centroid
The point in which the region will be perfectly balanced if suspended from that point
Centroid formula
Centroid formula
¼ of the height
Centroid of cone from the base is always
Location of Centroid of rectangle
Location of Centroid of Triangle
Location of Centroid of Spandrel
Area of rectangle
Area of triangle
Area of spandrel
Area of Parabola (Section)
Location of Centroid of Parabola (Section)
Location of Centroid of Semi-circle
Area of Semi-circle
Location of Centroid of Semi-ellipse
Area of Semi-ellipse
Location of centroid of hemisphere
Location of centroid of cone
Volume of cone
volume of hemisphere
Moment
Tendency to rotate
Inertia
Resistance to change
Mass moment of inertia (Resistance to rotation)
r=centroid
*works for a strip parallel to AOR
Area moment of inertia (Resistance to bending)
y=centroid
Higher area=more resistance to change
*works for a strip parallel to AOR
Average value of a function
Mean value theorem
