What is the definite integral of zero?
1 Answer
If you mean
This can be seen in a number of ways.
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Intuitively, the area under the graph of the null function is always zero, no matter over what interval we chose to evaluate it. Therefore,
#int_a^b 0 dx# should be equal to#0# , although this isn't an actual computation. -
Note the derivative of a constant function
#d/(dx)C=0# .
By the Fundamental Theorem of Calculus, we get
#int_a^b 0 dx = int_a^b d/(dx) C dx = C(b) - C(a) = C - C = 0# -
Consider the Riemann Sums of the function
#0# :
#sum_i^n f(x_i) Delta x_i = sum_i^n 0 Delta x_i ,#
where#Delta x_i# are the lengths of the divisions of the interval#[a,b]# .
No matter how we choose to divide the interval, this sum is always equal to#0# , since#0 Delta x_i=0# .
Therefore, the limit
#lim_(n to oo) sum_i^n 0 Delta x_i = int_a^b 0 dx = 0#
