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Application of Integral Calculation Rules
i-06-0A3a
number
200
randRange(1,10) randRangeNonZero(-10,10) function( t ) { return [t, abs(t)]; } function( t ) { return [t, 0]; } function( t ) { return [t, -t*t]; }

Let f an even function with

\displaystyle\int_{0}^{U} f(x)\; dx = I.

Calculate

\displaystyle\int_{0}^{-U} f(x) \; dx.

-I

We swap the bounds of \displaystyle\int_{\color{blue}{0}}^{ \color{red}{-U}} f(x)\; dx and get a negative sign

\displaystyle \int_{\color{blue}{0}}^{ \color{red}{-U}} f(x)\; dx = - \int_{\color{red}{-U}}^{\color{blue}{0}} f(x)\; dx.

The graph of an even function is symmetric with respect to the y-Achse.

Here we see two example graphs in the same coordinate system:

graphInit({ range: [[ -2.5, 2.5 ], [ -1, 2.5 ] ], scale: [ 30, 30 ], gridStep: [ 1, 1 ], tickStep: [ 3, 3 ], labelStep: [ 3, 3 ], unityLabels: true, }); // draw curve style({ stroke: BLUE, strokeWidth: 2.2 }, function() { plot( function( x ) { return abs (x); }, [ -2, 2 ] ); }); graphInit({ range: [[ -2.5, 2.5 ], [ -4.5, 2.5 ] ], scale: [ 30, 30 ], gridStep: [ 1, 1 ], tickStep: [ 3, 3 ], labelStep: [ 3, 3 ], unityLabels: true, }); // draw curve style({ stroke: ORANGE, strokeWidth: 2.2 }, function() { plot( function( x ) { return - x*x; }, [ -2, 2 ] ); });

The areas are each of equal size.

style({ fill: BLUE, stroke: BLUE, opacity: 0.5 }, function() { plotParametric( fn1, [-2, 2], 1, fn2); }); style({ fill: ORANGE, stroke: ORANGE, opacity: 0.5 }, function() { plotParametric( fn3, [-2, 2], 1, fn2); });

From the symmetric graph, we see:

\displaystyle \pm\left(\int_{\color{red}{-U}}^{\color{blue}{0}} f(x)\; dx\right) = Left area = Right area = \pm\left( \displaystyle \int_{0}^{U} f(x)\; dx\right) .

And together:

\displaystyle\int_{0}^{-U} f(x) \; dx = - \int_{\color{red}{-U}}^{\color{blue}{0}} f(x)\; dx = -I.