Question

Which of these integrals is equivalent to the given integral?

a) ∫a b f(x)dx = −∫b a f(x)dx
b) ∫a b f(x)dx = ∫b a f(−x)dx
c) ∫a b f(x)dx = ∫a b f(−x)dx
d) ∫a b f(x)dx = ∫−a−b f(x)dx

Answer 1

The integral that is equivalent to the given integral is option b) ∫a b f(x)dx = ∫b a f(-x)dx. In this integral, we integrate from b to a, but evaluate the function at -x, reflecting the curve about the y-axis.

Step-by-step explanation:

The integral that is equivalent to the given integral is option b) ∫a b f(x)dx = ∫b a f(-x)dx

To see why this is true, let's consider the limits of integration. In the original integral, we integrate from a to b, which means we are taking the area under the curve starting from a and ending at b. In the equivalent integral, we integrate from b to a, but instead of evaluating the function at x, we evaluate it at -x. This essentially reflects the curve about the y-axis. So, both integrals calculate the same area, but in opposite directions.