Integration and geometry
The integral $$\displaystyle \int_a^b f(x)\,dx$$ is the signed area under the graph $$y=f(x)$$; any area where the function is negative is subtracted.
This extends to computing general areas.
- Absolute area between the curve and the $$x$$-axis $$$ \int_a^b \vert f(x)\vert \,dx. $$$
- Signed area between two curves $$y=f(x)$$ and $$y=g(x)$$ $$$ \int_a^b \big(f(x)-g(x)\big) \,dx. $$$
The integral $$\displaystyle \int_a^b f(x)\,dx$$ is the area between the graph $$y=f(x)$$ and the $$x$$-axis.
The signed area between the curve, the $$y$$-axis , $$y=c$$ and $$y=d$$ is given by the integral $$$ \int_c^d f^{-1}(y) \,dy. $$$
The proof is simply that the graph of $$f^{-1}$$ is the graph of $$f$$ after swapping the $$x$$-axis and the $$y$$-axis.
The volume within a surface of revolution around the $$x$$-axis bounded by the curve $$y=f(x)$$ and planes $$x=a$$ and $$x=b$$ is given by the integral $$$ \pi \int_a^bf(x)^2\,dx. $$$
The formula for the volume arises from the fact that the figure can be sliced into cylinders with width $$dx$$ and radius $$f(x)$$. Each cylinder has volume $$\pi f(x)^2\,dx$$. These infinitesimal volumes add up to form the integral.