Functions Defined by Integrals

Part 1. The Function F(x) and Its Derivative

1. Define and plot the functions f and F.

f[x_] :=  Sqrt[25 - x^2];
Plot[f[x], {x, 0, 5},  
        AspectRatio -> 1];
F[x_] :=  
        Integrate[f[t], {t,  0,  x}];
Plot[Evaluate[F[x]], {x,  0,  5},  
        PlotStyle ->  RGBColor[0, 0, 1]];

2. Make a table of values of F[x]:  Vary x in the following command, and fill in lines of the table as you compute them.  To create additional rows, press Ctrl-Enter within the table.

x  =  1
N[F[x]]

({{x, F[x]}, {0.5, 2.49583}, {1.0, \[Placeholder]}})

3. Fill in the following table of approximate derivative values for F[x],and compare with corresponding values of f[x].

x  =  0.5
Delta  =  0.001
{x,  (F[x + Delta] - F[x])/Delta,  
      f[x]}

({{x, (F[x + Delta] - F[x])/Delta, f[x]}, {0.5, 4.97489, 4.97494}, {1.0, \[Placeholder], \[Placeholder]}, {1.5, \[Placeholder], \[Placeholder]}, {2.0, \[Placeholder], \[Placeholder]}, {2.5, \[Placeholder], \[Placeholder]}, {3.0, \[Placeholder], \[Placeholder]}, {3.5, \[Placeholder], \[Placeholder]}, {4.0, \[Placeholder], \[Placeholder]}, {4.5, \[Placeholder], \[Placeholder]}})

Answer steps 4 through 10 here:

11. The following commands will give a symbolic formula for F[x].

Clear[x];
F[x]

Now differentiate and simplify (if necessary).

D[F[x], x]  //  Simplify

Part 2. Integrating f'[x]

Define the function g[x].

Df[x_] :=  D[f[x], x];
g[x_] :=  Integrate[Df[t], {t, 0, x}];

Graph Df and g.

Plot[Evaluate[Df[x]], {x,  0, 5}];
plotg  =  
        Plot[Evaluate[g[x]], {x,  0,  5},  
          PlotStyle -> RGBColor[0, 0, 1],  
          AspectRatio -> 1];

2. Compare the graph of g with the graph of f.

plotf  =  
        Plot[f[x], {x,  0,  5},  
          AspectRatio -> 1];
Show[plotg, plotf,  AspectRatio -> 2];

Answers to questions 3, 4, and 5:

6. Compare the symbolic formula for g[x] with the definition of f[x].

g[x]

Answer to step 7:

Part 3. A Second Example

1-3. Define and plot f[x] and F[x].

f[x_] :=  Sin[x];
F[x_] :=  Integrate[f[t],  {t,  0,  x}];
Plot[f[x], {x, 0, Pi/2}];
Plot[Evaluate[F[x]], {x, 0, Pi/2},  
        PlotStyle ->  RGBColor[0, 0, 1]];

4. Define and plot g[x], and compare with the graph of f[x].

Df[x_] :=  D[f[x],  x];
g[x_] :=  
        Integrate[Df[t],  {t,  0,  x}];
Plot[Evaluate[g[x]], {x,  0,  Pi/2}, 
        PlotStyle ->  RGBColor[0, 0, 1]];
Plot[f[x], {x,  0,  Pi/2}];

Part 4. A Third Example

1. Define and plot f[x] and F[x].

f[x_] :=  Exp[-(x^2)/2];
F[x_] :=  
        Integrate[f[t],  {t,  0,  x}];
Plot[f[x], {x, 0, 3}];
Plot[Evaluate[F[x]], {x, 0, 3},  
        PlotStyle ->  RGBColor[0, 0, 1]];

2. Get the symbolic form of F[x].

F[x]

3. Differentiate and compare with the definition of f[x].

D[F[x], x]

4. Define g[x] and compare with f[x].

Df[x_] :=  D[f[x],  x];
g[x_] :=  
        Integrate[Df[t],  {t,  0,  x}];
g[x]

Part 5. The Fundamental Theorems of Calculus

Answers to the questions about FTC 1 and 2:

Part 6. Summary

Answers to the summary questions: