MATH115 GEOMETRY AND CALCULUS

Workshop Exercises 4

• Find the greatest and least distances from the origin of the curve $5x^2+4xy+2y^2=30$.

• Find the greatest and least values of $xy$ on the ellipse $4x^2-3xy+y^2=14$.

• Let $S$ be the region in the plane which is bounded by the curves $y=x(x-2)$ and $y=x$.

  • Find the points of intersection of the two curves and sketch the region $S$.

  • Divide the region $S$ into vertical strips (i.e. having fixed values of $x$) and hence evaluate the integral:

    $$\int {\int }_S\frac{y}{x^2}𝑑x𝑑y.$$

• Let $T$ be the triangle that is bounded by the lines $x=0$, $y=1$ and $y=x$.

  • Sketch $T$, and show that

    $$\int {\int }_T\frac{x^n}{y}𝑑x𝑑y={\int }_0^1{\int }_0^y\frac{x^n}{y}𝑑x𝑑y=\frac{1}{{(n+1)}^2}.$$

  • Write down the other repeated integral expression, and hence or otherwise find the value of

    $${\int }_0^1{x}^n\log xdx.$$

• Use polar coordinates to evaluate

  • ${\displaystyle \int {\int }_{A}y𝑑x𝑑y}$, where $A$ is the region defined by $y\ge 0$, $1\le x^2+y^2\le 4$.

  • ${\displaystyle \int {\int }_D\frac{1}{x^2+y^2+1}𝑑x𝑑y}$, where $D$ is the disc defined by $x^2+y^2\le 1$.

• Use appropriate changes of variables to determine the following integrals

  • $\int {\int }_R{x}^{2}y𝑑x𝑑y$, where $R$ is the region given by $1\le xy\le 2$, $\frac{1}{x^3}\le y^2\le \frac{2}{x^3}$.

  • $\int {\int }_S𝑑x𝑑y$, where $S$ is the region given by $0\le xy\le 3$, $1\le x^{2}y+x\le 2$.