Integrals

Premise (Relates heavily to Riemann Sums):

Imagine you have a car travelling a displacement $s(t)$, where $T$ is a specific time on his journey. He gradually speeds up, steadies out and slows down. His velocity at any one time $t$ is $v(t)=t(8-t)$.

A tiny change in displacement $s$ (I.E. $ds$) can be gotten using a tiny change in time (I.E. $dT$) multiplied by the velocity at that point in time $T$. Rearranged you get $\frac{ds}{dt}=v(T)$.

Cue the idea of Integration

Signs in Integrals

Properties / Rules of Integrals

• ${\sin }^2(x)$
• $\sin ({\cos }^{-1}(z))=\sqrt{1-z^2}$
• $\int \frac{1}{x}dx=\ln |x|$

Rules of Integrals:

Comparison Properties of Integrals: