To find the value F(x), we integrate the sine function from 0 to x. (a) To find F(π), we integrate sine from 0 to π: ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_18.png)
This means we're accumulating the weighted area between sin t and the t-axis from 0 to π: ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_graphik_2.png)
Evaluating the integral, we get ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_19.png)
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: ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_22.png)
On the graph, we're accumulating the weighted area between sin t and the t-axis from 0 to . ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_graphik_3.png)
The value 1 makes sense as an answer, because the weighted areas ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_24.png)
and ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_25.png) cancel each other out: ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_graphik_4.png)
We're left with one-half the area ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_26.png)
![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_graphik_5.png)
Since we found that ,
it makes sense for ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_28.png)
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. ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_32.png)
You can write the answer as ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_latek_33.png)
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: ![](https://media1.shmoop.com/images/calculus/calc_ftoc_secondft_graphik_6.png)
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. |