Questions: Give a geometrical explanation as to why ∫ from a to a f(x) dx = 0 Choose the correct answer A. Any function f(x) integrated from a to a will be zero because exactly half of the function will lie to the right of the y-axis, and the other half left of the y-axis B. The length of the interval [a, a] is a-a=0 Therefore, the net area of the region bounded between f(x) and the x-axis, within the limits of integration, must also be zero C. Any function f(x) integrated from a to a will be zero because exactly half of the function will lie above the x-axis, and the other half below the x-axis D. The height of the function on the interval [a, a] is f(a)-f(a)=0. Therefore, the net area between f(x) and the x-axis, within the limits of integration, must also be zero.

Solution Steps

To solve this problem, we need to understand the concept of definite integrals in terms of geometry. The integral of a function over an interval represents the net area between the function and the x-axis over that interval. If the interval has zero length, the net area is zero, regardless of the function's behavior.

The correct answer is B. The length of the interval \([a, a]\) is \(a-a=0\). Therefore, the net area of the region bounded between \(f(x)\) and the x-axis, within the limits of integration, must also be zero.

The integral \(\int_{a}^{a} f(x) \, dx\) represents the net area under the curve \(f(x)\) from \(x = a\) to \(x = a\). Since the interval \([a, a]\) has zero length, the net area is zero.

The length of the interval \([a, a]\) is calculated as \(a - a = 0\). This means there is no horizontal distance over which to accumulate any area under the curve.

Since the interval length is zero, the integral evaluates to zero regardless of the function \(f(x)\). This is because there is no space to accumulate any area, making the integral \(\int_{a}^{a} f(x) \, dx = 0\).

Final Answer