To find the value *F*(*x*), we integrate the sin function from 0 to *x*. (a) To find *F*(π), we integrate sin from 0 to π: This means we're accumulating the weighted area between sin *t* and the *t*-axis from 0 to π: Evaluating the integral, we get The value of *F*(π) is the weighted area between sin t and the horizontal axis from 0 to π, which is 2. (b) To find we substitute in for *x*: On the graph, we're accumulating the weighted area between sin *t* and the *t*-axis from 0 to . The value 1 makes sense as an answer, because the weighted areas and cancel each other out: We're left with one-half the area Since we found that , it makes sense for to be 1. (c) To find we put in for *x*. The upper limit of integration is less than the lower limit of integration 0, but that's okay. You can write the answer as if you want. On the graph, we're accumulating the weighted area between sin *t* and the *t*-axis from 0 to . This means we're integrating going left: Since we're accumulating area below the axis, but going left instead of right, it makes sense to get a positive number for an answer. |