How do you find the equation to an integral when the equation is not given but the answer to the integral is?

Question

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

Answer 1

The integral of 8f(x)-9 has two parts of area, say A₁₌ 8f(x) defined by variable x and A₂₌ -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,

Q₁:    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  ( A₁+  A₂)

          =38.5

Q₂:  Area from -8 to -0.5

        Variable area A₁= 8f(x) = 8*8=64;  

        Fixed area A₂ = -9x|_{-8}^{-0.5}= -9 (-0.5-(-8))= -67.5  

        Total area A = A_{1 +} A₂ = -3.5

Q₃:  Area from -8 to -5.5

        A₁  = 8*9 =72;    A₂ = -9x|_{-8}^{-5.5}= -9 (-5.5-(-8)) = -22.5:

        A  = A₁ + A2 = 49.5

Q₄:   Area from -3 to -0.5

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

        A = -14.5

Proof:  Q₂ = -Q₁ + Q₃ + Q₄    (Q₁ is negative because of up-down bound change)

         -3.5 = -38.5 + 49.5 - 14.5        

 

  

 

Answer 2

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