Area
$$\int_a^b \text{Top}(x)-\text{Bottom}(x)\,dx$$
Area
$$\int_c^d \text{Right}(y)-\text{Left}(y)\,dy$$
Horiz Axis
$$\int_a^b \Big(R(x)\Big)^2-\Big(r(x)\Big)^2\,dx$$
Vert Axis
$$\int_c^d \Big(R(y)\Big)^2-\Big(r(y)\Big)^2\,dy$$
⟂ to x-Axis
$$\int_a^b \text{Area}(x)\,dx$$
⟂ to y-Axis
$$\int_c^d \text{Area}(y)\,dy$$
Semicircle
$$\frac{\pi}{8}$$
Equilateral Triangle
$$\frac{\sqrt{3}}{4}$$
$$v(t)=x'(t)$$
$$x(t_2)=x(t_1)+\int_{t_1}^{t_2} v(t)\,dt$$
$$a(t)=v'(t)=x”(t)$$
$$v(t_2)=v(t_1)+\int_{t_1}^{t_2} a(t)\,dt$$
$$\text{Speed}=|v(t)|$$
$$\text{Distance}=\int_{t_1}^{t_2} |v(t)|\,dt$$
Average Rate of Change
$$\frac{f(b)-f(a)}{b-a}$$
Average Value
$$\frac{1}{b-a}\int_a^b f(x)\,dx$$