Let *f*(*x*) = sin *x* and *a* = 0. Define

Find each value and represent using a graph of sin *t*.

(a) *F*(-π)

(b) *F*(2π)

(c)

(d) Consider all real values of *x*. For what values of *x* is *F*(*x*)

(i) positive?

(ii) negative?

(iii) zero?

Answer

(a)

Because we're integrating to the left, this value is the negative of the weighted area between sin *t* and the *t*-axis on [-π,0]. So it's the positive value of this area:

(b)

This makes sense, because the weighted areas between sin *t* and the *t*-axis on [0,2π] cancel out:

(c)

Since we're integrating to the left, this is the negative of the weighted area between sin *t* and the *t*-axis on :

The weighted area on is zero, because the areas above and below the axis on this interval are equal:

We're left with the negative of the negatively weighted area between sin *t* and the axis on , which is this area:

(d) We're integrating starting at 0. When we integrate sin *t* on the interval [0,0] we get 0, so

When *x* is any integer multiple of 2π (positive or negative),

because the weighted areas the integral accumulates cancel each other out:

For any value of *x* in (0,π], the function

accumulates a positively weighted area, so the value of the function will be positive:

For any value of *x* in (π, 2π), the area below the *t*-axis is smaller than the area above the *t*-axis, so the function

is still positive:

Since , when we move *x* into the interval (2π, 3π] the values of *F*(*x*) are still positive:

And similarly for *x* in the interval (3π, 4π):

We can generalize to say that *F*(*x*) is zero when *x* is an integer multiple of 2π and positive for all positive values of *x* that aren't multiples of 2π.

When *x* is a negative multiple of 2π, we know that *F*(*x*) = 0. What about when *x* < 0 is not a multiple of 2π?

When *x* < 0 the function *F*(*x*) integrates to the left, so we count areas below the axis positively and areas above the axis negatively.

For *x* in [-π,0) the function *F*(*x*) counts this area positively. This means *F*(*x*) > 0 for *x* in [-π,0).

For *x* in (-2π,-π) the function *F*(*x*) considers the area below the axis positively, and the area above the axis negatively. Since there's more area below the axis than above the axis, *F*(*x*) is positive.

Generalizing, *F*(*x*) will be positive for all *x* < 0 that aren't multiples of 2π.

Putting all our observations together, we can finally answer the question.

(i) *F*(*x*) is positive for all *x* that aren't multiples of 2π.

(ii) *F*(*x*) is never negative.

(iii) *F*(*x*) is zero for all *x* that are multiples of 2π.

In the previous integral we used 0 as the lower limit of integration, but we can build a similar function using any lower limit of integration we want.