19fa-1910-recitation05-solutions.pdf - u00a76.1 A REA...

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§ 6.1 A REA BETWEEN CURVES § 6.2 S ETTING UP INTEGRALS N AME : S OLUTIONS Math 1910 A REA BETWEEN CURVES (1) The graph of x = f ( y ) is the graph of y = f ( x ) reflected across the line y = x . (2) The area between y = f ( x ) and y = g ( x ) from x = a to x = b is Z b a ( y top - y bottom ) dx . ( 1 ) (3) The area between x = g ( y ) and x = h ( y ) from y = a to y = b is Z b a x right - x left dx ( 2 ) I NTEGRALS AS AVERAGES (4) The average value of f ( x ) over the interval [ a , b ] is 1 b - a Z b a f ( x ) dx . ( 3 ) (5) The Mean Value Theorem for Integrals says that if f is continuous on [ a , b ] with mean value M , then there is some c [ a , b ] such that f ( c ) = M . V OLUME AS INTEGRAL OF CROSS - SECTIONAL AREAS (6) Pythagorean Theorem: If a right triangle has side lengths a , b and c , where c is the length of the hypotenuse, then a 2 + b 2 = c 2 . (7) Similar Triangles: If two triangles are similar (i.e. have angles of equal measure), then their side lengths are proportional. a b c d e f a d = b e = c f (8) If a shape has cross-sectional area A ( y ) and height extends from y = a to y = b , then it’s volume is Z b a A ( y ) dy . ( 4 ) (9) Cavilieri’s Principle says if two solids have equal cross-sectional areas, then they also have equal volumes. 1

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