Application of Integral Calculation Rules

de-CH

utf-8

math math-format

i-06-02

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 be an even function with

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

Calculate

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

I

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

Here we see two examples of a graph 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 symmettry of the 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).

Thus:

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