Final answer:
The integral of a continuous function that is equal to 0 everywhere from a to b is 0, because the area under its curve (a horizontal line at the x-axis) between any two points is nonexistent.
Step-by-step explanation:
To prove that the integral of a continuous function f(x) from a to b is 0 if f(x) = 0, we can use the fundamental concept of Riemann integration. The integral of a function over an interval can be thought of as the area under the curve of the function between the two bounds of the interval. For a continuous function that is equal to 0 everywhere, the graph is a horizontal line on the x-axis.
The area under the horizontal line f(x) = 0 from a to b is nonexistent, as it is simply the line segment on the x-axis between the points a and b. Since there is no area above the x-axis and below this horizontal line, the integral is equal to 0. This is true regardless of the values of a and b, provided that the function remains constant at 0 throughout the interval.