Winter 2016 – Q.6

6. a) Evaluate $latex \int_0^1\int_0^{1-x}\int_0^{1-x-y}xyz\;dz\;dy\;dx$ [6M]

We have,

$latex \int_0^1\int_0^{1-x}\int_0^{1-x-y}xyz\;dzdydx$

$latex \begin{array}{l}=\int_0^1\int_0^{1-x}xy\left[\int_0^{1-x-y}z\;dz\right]dydx\\\\=\int_0^1\int_0^{1-x}xy\left[\frac{z^2}2\right]_0^{1-x-y}dydx\\\\=\int_0^1\int_0^{1-x}\frac{xy\left(1-x-y\right)^2}2dydx\\\\=\frac12\int_0^1\int_0^{1-x}xy\left(1+x^2+y^2-2x-2y+2xy\right)dydx\\\\=\frac12\int_0^1\left[\int_0^{1-x}\left[xy+x^3y+xy^3-2x^2y-2xy^2+2x^2y^2\right]dy\right]dx\\\\=\frac12\int_0^1\left[x\left\{\frac y2\right\}_0^{1-x}x^3\left\{\frac{y^2}2\right\}_0^{1-x}+x\left\{\frac{y^4}4\right\}_0^{1-x}-2x^2\left\{\frac{y^2}2\right\}_0^{1-x}-2x\left\{\frac{y^3}3\right\}_0^{1-x}+2x^2\left\{\frac{y^3}3\right\}_0^{1-x}\right]dx\end{array}$

$latex =\frac1{24}\int_0^1\left[6x\left(1-2x+x^2\right)+6x^3\left(1-2x+x^2\right)+3x\left(1+x^4-4x^3+6x^2-4x\right)-12x^2\left(1-2x+x^2\right)-8x\left(1-x^3+3x^2-3x\right)+8x^2\left(1-x^3+3x^2-3x\right)\right]dx$

b) Find the mass of area bounded by the curves $latex y=x^2\;and\;x=y^2;$, if the density at any point is $latex \rho=\lambda\left(x^2+y^2\right)$. [6M]

The point of intersection of the two parabolas are given by solving the given equations

y = x² and x = y²

y = x² = (y²)² = y⁴

y⁴ = y

∴ $latex \frac{y^4}y=y^3=1$

∴ y = 1

similarly x = 1

Required mass $latex \int_A\int e\;dxdy$

$latex =\int_{y=0}^{1}\int_{x=y^{2}}^{\sqrt{y}}\lambda \left ( x^{2}+y^{2}\right ) dx dy$

$latex =\lambda \int_{0}^{1}\left [ \int_{y^{2}}^{\sqrt{y}}\left ( x^{2}+y^{2} \right )dx \right ]dy$

$latex =\lambda \int_{0}^{1}\left [ \frac{x^{3}}{3}+y^{2}x\right ]_{y^{2}}^{\sqrt{y}}dy$

$latex \begin{array}{l}=\lambda\int_0^1\left[\frac{y^{\displaystyle\frac32}}3+y^\frac52-\frac{y^6}3-y^4\right]dy\\\\=\lambda\int_0^1\left[\frac{y^{\displaystyle\frac32}}3-\frac{y^6}3+y^\frac52-y^4\right]dy\\\\=\lambda\;\left[\frac{2y^{\displaystyle\frac52}}{15}-\frac{y^7}{21}+\frac{2y^\frac72}7-\frac{y^5}5\right]_0^1\\\\=\lambda\;\left[\frac2{15}-\frac1{21}+\frac27-\frac15\right]\\\\=\lambda\;\frac{\left(14-5+30-21\right)}{105}\\\\=\frac{18\lambda}{105}\\\\=\frac{6\lambda}{35}\end{array}$

∴ Mass of area $latex =\frac{6\lambda}{35}$

c) Evaluate $latex \iint\frac{rdrd\theta}{\sqrt{a^2+r^2}}$ over one loop of the lemniscate $latex r^2=a^2cos2\theta$. [6M]