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Chapter 6 | Applications of Integration
• Work done on an object W = ∫ a b F ( x ) dx • Hydrostatic force on a plate F = ∫ a b ρw ( x ) s ( x ) dx • Mass of a lamina m = ρ ∫ f ( x ) dx
a b
• Moments of a lamina M x = ρ ∫ a b ⎡ ⎣ f ( x ) ⎤ ⎦ 2 2
dx and M y = ρ ∫ a b
xf ( x ) dx
• Center of mass of a lamina x – = M y m and y – = M x m • Natural logarithm function • ln x = ∫ 1 x 1 t dt Z • Exponential function y = e x • ln y = ln( e x ) = x Z
KEY CONCEPTS 6.1 Areas between Curves • Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. • To find the area between two curves defined by functions, integrate the difference of the functions. • If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals and add the results to find the area of the region. • Sometimes it can be easier to integrate with respect to y to find the area. The principles are the same regardless of which variable is used as the variable of integration. 6.2 Determining Volumes by Slicing • Definite integrals can be used to find the volumes of solids. Using the slicing method, we can find a volume by integrating the cross-sectional area. • For solids of revolution, the volume slices are often disks and the cross-sections are circles. The method of disks involves applying the method of slicing in the particular case in which the cross-sections are circles, and using the formula for the area of a circle. • If a solid of revolution has a cavity in the center, the volume slices are washers. With the method of washers, the area of the inner circle is subtracted from the area of the outer circle before integrating. 6.3 Volumes of Revolution: Cylindrical Shells • The method of cylindrical shells is another method for using a definite integral to calculate the volume of a solid of revolution. This method is sometimes preferable to either the method of disks or the method of washers because we integrate with respect to the other variable. In some cases, one integral is substantially more complicated than the
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