int; integral sign

Find int (bottom: -3 top: -5.5) (8*f(x)-9)dx when int (bottom:-5.5 top:-3) f(x)dx = -2

Also; let

int (bottom: -8 top: -0.5) f(x)dx =8

int (bottom: -8 top: -5.5) f(x)dx = 9

int (bottom: -3 top: -0.5) f(x)dx = 1

int; integral sign

Find int (bottom: -3 top: -5.5) (8*f(x)-9)dx when int (bottom:-5.5 top:-3) f(x)dx = -2

Also; let

int (bottom: -8 top: -0.5) f(x)dx =8

int (bottom: -8 top: -5.5) f(x)dx = 9

int (bottom: -3 top: -0.5) f(x)dx = 1

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The integral of 8f(x)-9 has two parts of area, say A_{1=} 8f(x) defined by variable x and A_{2=} -9 defined by constant number -9. We take integral to get the area from -3 to -5.5 by changing the up-down bound order from -5.5 to -3 and the sign + to -. Then, proceed all the rest as,

Q_{1}: Area from -3 to -5.5

\int_{-3}^{-5.5} ( 8f(x) - 9) dx

= -\int_{-5.5}^{-3} ( 8 f(x) - 9) dx

= -8 \int_{-5.5}^{-3} f(x) dx + 9 \int_ {-5.5}^{-3} dx

= (-8)(-2) + 9 \int_{-5.5}^{-3} dx

=-16 + 9x|_{-5.5}^{-3}

=16 + (-27-(-49.5))

=16 + 22.5 ( A_{1}+ A_{2})

=38.5

Q_{2}: Area from -8 to -0.5

Variable area A_{1}= 8f(x) = 8*8=64;

Fixed area A_{2 }= -9x|_{-8}^{-0.5}= -9 (-0.5-(-8))= -67.5

Total area A = A_{1 + }A_{2 }= -3.5

Q_{3}: Area from -8 to -5.5

A_{1 } = 8*9 =72; A_{2 }= -9x|_{-8}^{-5.5}= -9 (-5.5-(-8)) = -22.5:

A = A_{1} + A2 = 49.5

Q_{4}: Area from -3 to -0.5

A1 = 8*1 = 8; A2 = -9( -0.5 - ( -3))= -22.5

A = -14.5

Proof: Q_{2} = -Q_{1} + Q_{3} + Q_{4 } (Q_{1} is negative because of up-down bound change)

-3.5 = -38.5 + 49.5 - 14.5

George C. | Humboldt State and Georgetown graduateHumboldt State and Georgetown graduate

This is more a linear algebra problem than a calculus problem. Set up a matrix from your given information.

Let f(-.5)=a. f(-8)=b, f(5.5)=c, f(-3)=d

From a plot, these are not a straight line. Their arrangement resembles a cubic function. Thus the equation ax^3 + bx^2 + cx + d will be the form. Using the coefficients as the unknowns and the values of x as the coefficients, the matrix becomes

a -b =8

-b +c =9

a -d =1

-c +d =2

-.125 .25 -.5 1 8

-512 64 -8 1 9

-166.375 30.25 -5.5 1 1

-27 9 -3 1 -2

Huffing and puffing gives

f(x) = .0853 x^3 + 1.808 x^2 + 9.4107 x + 12.264 f'x)= .2559 x^2 + 3.6160 x + 9.4107

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