Chapter 6 | Applications of Integration
757
surface area
symmetry principle the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces the symmetry principle states that if a region R is symmetric about a line l , then the centroid of R lies on l this theorem states that the volume of a solid of revolution formed by revolving a region around an external axis is equal to the area of the region multiplied by the distance traveled by the centroid of the region a special case of the slicing method used with solids of revolution when the slices are washers the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance KEY EQUATIONS • Area between two curves, integrating on the x -axis A = ∫ theorem of Pappus for volume washer method work ⎡ ⎣ f ( x )− g ( x ) ⎤ ⎦ dx
a b
• Area between two curves, integrating on the y -axis A = ∫ c d ⎡ ⎣ u ( y )− v ( y ) ⎤ ⎦ dy • Disk Method along the x -axis V = ∫ a b π ⎡ ⎣ f ( x ) ⎤ ⎦ 2 dx • Disk Method along the y -axis V = ∫ c d π ⎡ ⎣ g ( y ) ⎤ ⎦ 2 dy • Washer Method V = ∫ a b π ⎡ ⎣ ⎛ ⎝ f ( x ) ⎞ ⎠ 2 − ⎛ ⎝ g ( x ) ⎞ ⎠ 2 ⎤ ⎦ dx • Method of Cylindrical Shells V = ∫ a b ⎛ ⎝ 2 πxf ( x ) ⎞ ⎠ dx • Arc Length of a Function of x Arc Length = ∫ a b 1+ ⎡ ⎣ f ′( x ) ⎤ ⎦ 2 dx • Arc Length of a Function of y Arc Length = ∫ c d 1+ ⎡ ⎣ g ′( y ) ⎤ ⎦ 2 dy • Surface Area of a Function of x Surface Area = ∫ a b ⎛ ⎝ 2 πf ( x ) 1+ ⎛ ⎝ f ′( x ) ⎞ ⎠ 2 ⎞ ⎠ dx • Mass of a one-dimensional object m = ∫ a b ρ ( x ) dx • Mass of a circular object m = ∫ 0 r 2 πxρ ( x ) dx
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