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Page 1 of 9 DEFINITION Area Between Curves If f and g are continuous with ( ) ( ) between the curves f ( x ) and b, f x g x on the interval [, ] g( x ) from a to b is the integral of [ f ] a b, then the area g from a to b a [ ( ) ( )] A = f x g x dx. f ( x ) is the upper curve and g( x) is the lower curve. c d [ ( ) ( )] A = f y g y dy In this case, f ( y ) is the curve to the right and g( y) is the curve to the left. DEFINITION Volumes of Solids with Known Cross Sections For cross sections of area A( x ) taken perpendicular to the x-axis, the volume is the accumulation of the cross sections from a to b. V = b a A( x) dx. In order to find the volume, find the area of one cross section and accumulate the cross sections through integration.
Page of 9 KNOWN CROSS-SECTIONS We can accumulate the cross-sections if we can visualize one area. The thickness is dx. We accumulate through integration. Picture the side of the shape in the xy-plane perpendicular to the given axis. This side will be referred to as f ( x) g( x), assuming we have two functions and are finding the upper curve minus the lower curve. Square ( f g) π 8 f g Semi-circle ( ) 4 Equilateral Triangle ( ) f g Rectangle with the base in the xy-plane, perpendicular to the axis, and the height is 6 times the base. ( ( )) ( f g) 6 f g 6( f g) b π f g d a 8 x. The final integral for the semi-circle would be ( )
Page of 9 1. What is the area of the region enclosed by the graphs of f ( x) = x and gx= ( ) 4? (A) (B) 0 (C) 16 (D) (E) 64. The area of the region enclosed by the curve x = 4 and x = 5 is 1 g( x) = x, the x-axis, and the lines (A) 0 (B) ln (C) 5 ln 4 (D) ln (E) ln 6.GC What is the area between gx ( ) sin x = e and ( ) 0, π? hx = in the interval over [ ] (A) 0.406 (B) 0.78 (C) 1.551 (D).07 (E) 6.08
Page 4 of 9 4. What is the area of the region bounded by f ( x) g x =, and the line x = 1? x = e, ( ) 1 (A) (B) (C) (D) (E) e 1 e 1 1 e 1 e 1 4 e 5. What is the volume of the solid generated by revolving the curve y = x about x-axis from x = 0 to x = 4? (A) π (B) 4π (C) 16 π (D) 8π (E) π 6. What is the volume of the solid generated by the region between x =, and the x-axis revolved about the y-axis? y = x, the line (A) 7 π (B) 45 π (C) 81 π (D) 9π (E) 18π
Page 5 of 9 7.GC What is the area of the region enclosed by the function hx ( ) = xsinxand the π, π? x-axis over the interval [ ] (A).141 (B) 4.8 (C) 6.8 (D) 8.764 (E) 19.79 8.GC Rotate the ellipse generated? (A) 96 (B) 59.17 (C) 100.50 (D) 150.796 (E) 01.59 9x + y =6 about the x-axis. What is the volume of the solid 9. Let R be the region bounded by the function f( x) = ( x+ 1), the x-axis, the y-axis, and the line x = 7. The volume of the solid generated when R is revolved about the x-axis is given by 7 (A) ( x + 1) 0 7 (B) π ( x + 1) 0 dx dx ( ) 7 (C) π ( x + 1) 0 ( ) (D) π ( x + 1) 0 ( ) 7 (E) π ( x + 1 ) 1 dx dx dx
Page 6 of 9 10.GC What is the area of the region enclosed by the functions x hx ( ) = x + 1? gx ( ) x x = and (A) 0.099 (B) 0.647 (C) 1.95 (D).141 (E) 4.070 11. The region formed by the line x+ y =, the x-axis, and the y-axis is the base for a solid. The cross-sections perpendicular to the x-axis are squares, with one side of the square in the x-y plane. What is the volume of the solid? (A) 7 1 (B) (C) 4 (D) 1 (E) 8 1.GC The region formed by the line x+ y =, the x-axis, and the y-axis is the base for a solid. The cross-sections perpendicular to the x-axis are equilateral triangles, with one side of the triangle in the x-y plane. What is the volume of the solid? (A) 0.5 (B) 0.88 (C) 0.4 (D) 0.577 (E) 0.685
Page 7 of 9 Free Response Question 1. (Calculator allowed.) R The graph above shows the two functions f ( x) sin x x = e and gx ( ) e 1 =. (a) What is the area of region R, which is bounded by f(x), g(x), and the y-axis? (b) What is the volume of the solid generated when region R is revolved about the x- axis? (c) The region R is the base of a solid. Cross-sections perpendicular to the x-axis are squares. What is the volume of the solid?
Page 8 of 9 Free Response. (Calculator allowed.) The functions f( x) cos x = e and ( ) g x = are pictured in the graph above. (a) What is the area of the region R, bounded by f(x) above and g(x) below? (b) What is the volume of the solid generated when the region R is revolved about the x-axis? (c) What is the volume of the solid generated when the region R is revolved about the line? y = 1
Page 9 of 9 Free Response. (Calculator allowed.) The functions gx ( ) cos x = e and hx ( ) ln = xare graphed above. (a) What is the area of region R in the graph above? (b) What is the volume of the solid generated when R is revolved about the x-axis? (c) What is the volume of the solid generated when S is revolved about the line y =?