Multiple integrals and vector calculus 1: basic multiple integrals

5. Orthogonal circles

Two circles have equations (i) $x^2 + y^2 + 2ax + 2by + c = 0$ and (ii) $x^2 + y^2 + 2a'x + 2b'y + c' = 0$. Show that these circles are orthogonal for $2aa' + 2bb' = c + c'$.

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7. Computations of masses and inertia

a) A mass distribution in the positive $x$ region of the $xy$-plane and in the shape of a semicircle of radius $a$, centered on the origin, has mass per unit area $k$. Find using plane polar coordinates:

• Its mass $M$
• The coordinates $(\overline{x}, \overline{y})$ of its center of mass
• Its moments of inertia about the $x$ and $y$ axes
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b) Do as above for a semi-infinite sheet with mass per unit area $\rho = ke^{-(x^2 + y^2)/a^2}$

for $x \geq 0$ and $\rho = 0$ for $x < 0$, and where $a$ is constant.

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c) Evaluate the following integral: $\int_0^a \int_0^{\sqrt{a^2 - y^2}} (x^2 + y^2) \tan^{-1}(y/x) \, \de{x} \, \de{y}.$

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