If f(x) is continuous for all x, which of the following integrals necessarily have the same value?
b
I. ∫ f(x) dx
a
b
II. ∫ |f(x)| dx
a
b-c
III. ∫ f(x+c) dx
a-c
b
IV. ∫ (f(x)+c) dx
a
I and II only
I and III only
I, II and IV only
II, III, and IV only
not II. Consider f(x) = ∫[0,1] x^2-1
not IV. If F(x) = ∫f(x) dx then
∫(f(x)+c) dx = ∫f(x) dx + cx
Now, for III, let u = x+c
Then you have
∫[a,b] f(u) du
Makes sense, since f(x+c) is shifted to the left by c, and so is the interval of integration.
I and III only
The correct answer is "I and III only."
Explanation:
I. The integral of f(x) over the interval [a, b] is dependent on the function itself and the interval of integration. Since f(x) is continuous over all x, its integral will be well-defined on [a, b].
III. The integral of f(x + c) over the interval [a - c, b - c] is equivalent to the integral of f(x) over the interval [a, b], since the function f(x) is shifted horizontally by c units. Shifting the function horizontally does not affect the area under the curve, so the integral remains the same.
II. The integral of |f(x)| over [a, b] represents the area between the curve of f(x) and the x-axis, regardless of the sign of f(x). Since |f(x)| can change sign, the integral of |f(x)| may have a different value compared to the integral of f(x).
IV. The integral of (f(x) + c) over [a, b] represents the area between the curve of f(x) + c and the x-axis. Adding a constant c to the function f(x) will shift the entire curve vertically by c units. Shifting the curve vertically changes the overall area under the curve, so the integral will not necessarily have the same value as the integral of f(x).
To determine which of the integrals necessarily have the same value, we can use the properties of integrals and continuity of the function.
Let's break down each option to see which ones are correct:
I. ∫ f(x) dx / a to b
This is a simple integral of the function f(x) from point a to point b.
II. ∫ |f(x)| dx / a to b
This integral finds the area between the x-axis and the absolute value of the function f(x) over the interval [a, b].
III. ∫ f(x+c) dx / a-c to b-c
This integral represents the function f(x) translated to the left by c units. In other words, we shift the function horizontally by c units.
IV. ∫ (f(x)+c) dx / a to b
This integral represents the function f(x) shifted vertically upwards by c units. In other words, we add a constant c to the function.
Now, let's analyze the options:
I and II only: If a function is continuous, integrating its absolute value will give the same result as integrating the function itself. Therefore, Option I and Option II will necessarily have the same value.
I and III only: Option I represents the integral of f(x) as is, while Option III represents the integral of f(x+c), which is the same function shifted to the left by c units. These two integrals will generally have different values, so this option is incorrect.
I, II, and IV only: Similar to Option II, if we add a constant c to the function f(x) and integrate, the result will be the same as integrating the original function without the constant. Therefore, Option IV will have the same value as Option I and Option II. Hence, this option is correct.
II, III, and IV only: Option III represents shifting the function horizontally, and Option IV represents shifting the function vertically. These two options will generally result in different integrals, so this option is incorrect.
Based on our analysis, the correct answer is:
I, II, and IV only