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Review-and-Study-Guide

# Review-and-Study-Guide - Math 152 summary and review guide...

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Math 152 summary and review guide 5.1,5.2,6.5 Area and distance, Riemann sum, Definite integral, Average Write a definite integral as a Riemann sum and vice versa. Interpret an integral as a (signed) area under a curve, or accumulated distance given the velocity. Use formulas for n i =1 i and n i =1 i 2 to com- pute integrals (p.396). Understand basic and comparison properties (p.373 and p.375). Average value of f over [ a, b ] is 1 b a integraltext b a f ( x ) dx . By the Mean Value theorem this average equals f ( c ) for some c ( a, b ) if f is continuous. 5.3 Fundamental theorem of calculus (FTC) Roughly, if F ( x ) = f ( x ) then F ( x ) = integraltext x a f ( t ) dt and integraltext b a f ( x ) dx = F ( b ) - F ( a ). Use with chain rule to compute d dx integraltext g ( x ) f ( x ) h ( t ) dt = h ( g ( x )) g ( x ) - h ( f ( x )) f ( x )+ C . 5.4 Indefinite integrals. integraltext f ( x ) dx = F ( x ) + C means that F ( x ) - integraltext x a f ( t ) dt is a constant function, where the constant depends on the value of a . Memorize the table on p.392 except for the formulas in- volving csc x . The “Net change theorem” (p.394) is just the FTC. “Displacement” versus “Distance” (p.395). Power is the time deriv- ative of energy P ( t ) = E ( t ) (p.396). 5.5 Substitution rule This is the chain rule in reverse (p.401). Look for the form integraltext b a f ( g ( x )) g ( x ) dx and adjust the limits of integration (p.404). Symmetric functions sometimes integrate easier (p.405). 6.1, 10.4 Area between curves in Cartesian and polar coordinates Strategy: Sketch curves, and decide whether to integrate with respect to x or y . Find intersection points, and decide where f ( x ) g ( x ) to compute Area = integraltext b a | f ( x ) - g ( x ) | dx . Practice polar plotting of r = f ( θ ), and use A = integraltext b a 1 2 r 2 . Be aware of intersection points that arise from the facts ( r, θ ) = ( - r, θ + π ) and (0 , θ ) = (0 , θ ) (p.651). 6.2, 6.3 Volume of a solid of revolution by disk, shell and direct methods Strategy: Sketch and visualize the solid. Decide between washers or cylindrical shells. For y = f ( x ), washers are better around the x -axis and cylinders are better around the y -axis. Find intersec- tion points for limits of integration. For washers around x -axis, use V = integraltext b a A ( x ) dx = integraltext b a π ([ f ( x )] 2 - [ g ( x )] 2 ) dx (p.426). For cylinders around y -axis, use V = integraltext b a A ( x ) dx = integraltext b a 2 πxf ( x ) dy (p.434). For other solids, decide on a “slicing strategy”, plot the object so that the area of the “slice at x ”, A ( x ), is a simple function, and find the limits of integration for V = integraltext b a A ( x ) dx (p.429).

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Review-and-Study-Guide - Math 152 summary and review guide...

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