What are the symmetry properties of finding areas using integrals?
Symmetries can be used to simplify computation of definite integrals. Let us look at the following examples.
Example 1 (Even Function)
#int_{1}^1(3x^2+1) dx =2int_0^1(3x^2+1) dx=2[x^3+x]_0^1=2(20)=4#
Example 2 (Odd Function)
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The symmetry properties of finding areas using integrals are:

Symmetry about the xaxis: If a function ( f(x) ) is even (symmetric about the yaxis), then the area between the curve and the xaxis from ( x = a ) to ( x = a ) is twice the area from ( x = 0 ) to ( x = a ).

Symmetry about the yaxis: If a function ( f(x) ) is odd (symmetric about the origin), the area between the curve and the xaxis from ( x = a ) to ( x = a ) is zero.

Symmetry about the origin: If a function ( f(x) ) is symmetric about the origin, then the area between the curve and the xaxis from ( x = a ) to ( x = a ) is twice the area from ( x = 0 ) to ( x = a ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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