MAT 232 WA1 doc.docx - Written Assignment 1 Answer all...

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Written Assignment 1 Answer all assigned exercises, and show all work. Each exercise is worth 5 points. Section 5.2 2. Find the volume of the solid with cross-sectional area A ( x ). 0.01 ( ) 10 , 0 10 x A x e x ( ) b a V A x dx 10 0.01 0 10( ) x e dx 10 0.01 10 0.01 0 0 10 1000 0.01 x x e e 0.01(10) 0.01(0) 0.1 0 1000 1000 1000 1000 e e e e 1105.17 1000 105.17 105.17 V 6. Find the volume of a pyramid of height 160 feet that has a square base of side 300 feet. These dimensions are half those of the pyramid in example 2.1. How does the volume compare? Height=160ft Square base of side=300ft x= height above ground At x=0, cross-section= 300 At x=160, cross-section=0 f(x)= side length of square cross at height x f(0)=600, f(160)=0, f(x)=linear function slope(m) 300 0 300 0 160 160  15 ( ) 300 8 f x x  The cross-sectional area is simply the square of f(x) so that we have: WA 1, p. 1
160 160 2 0 0 15 ( ) ( 300) 8 x A x dx dx Evaluate the integral by substitution 15 300 8 u x  so that 15 8 du dx  160 0 2 2 0 300 15 8 ( 300) 8 15 V dx u du  300 300 3 3 3 2 0 0 8 8 (300) (0) 15 15 3 3 3 u V u du 3 8 9000000 15 4,800,000 V V ft 3 3 38,400,000 8 4,800,000 1 ft ft The new pyramid is 1/8 the volume of the original pyramid in example 2.1 10. A dome “twice as big” as that of exercise 9 (see text) has outline 2 120 120 x y for 120 120 x (units of feet). Find its volume. 120 2 0 V x dx 120 0 120 120 y dy 120 2 2 2 0 2 2 3 120 120 120 120 120 2 2 240 120 120 120 2 2 2 y y V= 864000π ft 3 12. A pottery jar has circular cross sections of radius 2 4 sin x inches for 0 2 . x Sketch a picture of the jar and compute its volume. ( ) b a V A x dx WA 1, p. 2
2 2 0 4 sin 2 x dx 2 2 0 16 8sin sin 2 2 2 2 sin 2 0 0 sin 0 16(2 ) 16cos 16(0) 16cos 2 2 2 2 2 2 x x dx 32 16 0 0 16 0 0 V=33π 3 inch 3 18. Compute the volume of the solid formed by revolving the region bounded by 2 2 1 2 , 4 y x y x   about (a) the x -axis; (b) y =4 Intersection of the curves: y 2 = y 1 WA 1, p. 3 y =x 2 y = 4-x 2 x-axis y=4
2 2 2 2 4 4 2 2 2 x x x x x (a) Revolved about the x-axis 2 2 2 2 2 2 2 2 2 2 ( ) (4 ) ( ) o i V r dx r r dx x x dx 2 2 4 4 2 3 3 2 3 2 (16 8 ) ( ) 8 8( 2 ) 8( 2 ) 16 16 2 16( 2) 3 3 3 x x x dx x x 96 2 32 2 3 3 64 2 3 V (b) Revolved about y = 4 2 2 2 2 2 2 2 2 2 2 ( ) (4 ) ( ) o i V r dx r r dx x x dx 2 2 4 4 2 3 3 2 3 2 (16 8 ) ( ) 8 8( 2 ) 8( 2 ) 16 16 2 16( 2) 3 3 3 x x x dx x x 96 2 32 2 3 3 64 2 3 V 20. Compute the volume of the solid formed by revolving the region bounded by 2 y x and 2 x y about (a) the y -axis; (b) x = 1. WA 1, p. 4
The intersection of two curves are: 2 2 0 (1 ) 0 0, 1 x x x x x x x x (a) Revolved around the y -axis 1 2 0 1 1 2 2 2 0 0 1 1 4 0 0 1 2 5 2 5 0 ( ) ( ) ( ) (1) (1) 2 5 2 5 5 2 3 10 10 10 3 10 o i V r r dy y dy y dy y dy y dy y y V

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