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# Integration and geometry

## Signed area below the graph of a function

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 $y=g(x)$ $$\int_a^b \big(f(x)-g(x)\big) \,dx.$$

Signed area under the graph of a function (left), and the area between two functions (right).

## Area on the left of the graph of a function

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.

Area between $y$-axis and curve

## Volume of surfaces of revolution

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.

Volume of revolution surface