Quick questions on Integration and its applications - TCE Mathematics Methods (Tasmania)

A few algebraic properties make definite integrals easier to handle and are regularly tested. Reversing the limits changes the sign, $\displaystyle\int_{a}^{b} f = -\int_{b}^{a} f$; an integral over a zero-width interval is zero, $\displaystyle\int_{a}^{a} f = 0$; a constant factor pulls out, $\displaystyle\int_{a}^{b} kf = k\int_{a}^{b} f$; a sum splits, $\displaystyle\int_{a}^{b}(f + g) = \int_{a}^{b} f + \int_{a}^{b} g$; and an interval can be split at any interior point, $\displaystyle\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$. The last property is exactly what justifies splitting at axis-crossings or curve-crossings.

A common application gives you a rate or a gradient function together with one known point, and asks for the original function. Integrate to introduce the constant $C$, then use the point to solve for it. This "boundary condition" technique is the same one that fixes the constants in kinematics, and it turns an indefinite integral into a single definite answer.

Because integration reverses differentiation, integrating a rate of change recovers the total change. If $\dfrac{dV}{dt}$ is the rate at which a tank fills, then $\displaystyle\int_{t_1}^{t_2}\frac{dV}{dt}\,dt$ is the volume added between $t_1$ and $t_2$.