two curves have equations y=x^2-4x+7 and y=6x-x^2-1. Find the area between the two curves
two curves have equations y=x^2-4x+7 and y=6x-x^2-1.
Find the area between the two curves
f(x)=6x−x2−1g(x)=x2−4x+7
intersections:
g(x)=f(x)x2−4x+7=6x−x2−1…2x2−10x+8=0|:2x2−5x+4=0(x−1)(x−4)=0x1=1andx2=4
area between the two curves:
4∫1( f(x)−g(x) ) dx=4∫1( 6x−x2−1−(x2−4x+7) ) dx=4∫1( 6x−x2−1−x2+4x−7 ) dx…=4∫1( 10x−2x2−8 ) dx=[ 10x22−2x33−8x ]41=[ 5x2−23x3−8x ]41=[ 5⋅(4)2−23(4)3−8⋅4 ]−[ 5(1)2−23(1)3−8⋅1 ]=5⋅16−23⋅64−32−5+23+8=51−23⋅64+23=51−23(64−1)=51−23(63)=51−2⋅21=9
The area between the two curves is 9
two curves have equations y=x^2-4x+7 and y=6x-x^2-1.
Find the area between the two curves
f(x)=6x−x2−1g(x)=x2−4x+7
intersections:
g(x)=f(x)x2−4x+7=6x−x2−1…2x2−10x+8=0|:2x2−5x+4=0(x−1)(x−4)=0x1=1andx2=4
area between the two curves:
4∫1( f(x)−g(x) ) dx=4∫1( 6x−x2−1−(x2−4x+7) ) dx=4∫1( 6x−x2−1−x2+4x−7 ) dx…=4∫1( 10x−2x2−8 ) dx=[ 10x22−2x33−8x ]41=[ 5x2−23x3−8x ]41=[ 5⋅(4)2−23(4)3−8⋅4 ]−[ 5(1)2−23(1)3−8⋅1 ]=5⋅16−23⋅64−32−5+23+8=51−23⋅64+23=51−23(64−1)=51−23(63)=51−2⋅21=9
The area between the two curves is 9