Some Totally Unremarkable Problems With Integrals

Some problems to do, right now, on a separate sheet of paper. Draw pictures for all of these problems.

1. Using our formula about how integrals involve antiderivatives and subtraction (the Fundamental Theorem of Calculus), calculate: \[ \int_5^{12} \!x^2\,dx\] Then calculate: \[ \int_5^{12} \!-x^2\,dx\] Compare and contemplate your results. (Note that I am not asking you here to find a certain area—I am asking you to calculate an integral.)
2. Likewise, calculate both: \[\int_0^\pi \!\sin(x)\,dx \quad\quad\text{and}\quad\quad \int_\pi^{2\pi} \!\sin(x)\,dx\] Then calculate: \[\int_0^{2\pi} \!\sin(x)\,dx\]
3. Make a conjecture¹ about the relationship between: \[ \int_a^b\! f(x)\,dx \quad\quad\text{and}\quad\quad \int_a^b \!-f(x)\,dx\] Then calculate both: \[\int_{12}^5 \!x^2\,dx \quad\quad\text{and}\quad\quad \int_\pi^0 \!\sin(x)\,dx\]
4. How are these different from the above problems? Make a conjecture about the relationship between: \[ \int_a^b \!f(x)\,dx \quad\quad\text{and}\quad\quad \int_b^a \!f(x)\,dx\]