Additional exercises 4

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Exercise

Evaluate the multiple integral $I={\iint }_{R}f(x,y)dxdy$ by iterated integration, where:

1. $R=[0,1]\times [0,1]$, $f(x,y)=27xy(x+y)$
2. $R=[0,1]\times [0,1]$, $f(x,y)=4x^3$
3. $R=[0,\pi /2]\times [0,\pi /2]$, $f(x,y)=7\sin (x+y)$.
4. $R=[0,1]\times [1,2]$, $f(x,y)=x{e}^{xy}$
5. $R=[0,1]\times [1,2]$, $f(x,y)=x{e}^{x-y}$

Exercise

Let $R=[0,\pi ]\times [0,\pi ]$.

Evaluate the following integral:

$${\iint }_R{(\sin x\sin y)}^{2}dxdy$$

Hint: integrals of the form ${\iint }_{R}f(x)g(y)dxdy$ are equal to the product of integrals.

Exercise

Let $S\subset {\mathbb{R}}^2$ be the triangular region with vertices $(0,0)$, $(\pi ,0)$, $(\pi ,\pi )$. Evaluate integral

$${\iint }_{S}x\cos (x+y)dxdy.$$

Exercise

Let $S\subset {\mathbb{R}}^2$ be the region bounded by the curves $y=\sin x$ and the line segment $\{(x,0):x\in [0,\pi ]\}$.

Sketch $S$ and evaluate the integral

$${\iint }_S(x^2-y^2)dxdy.$$

Exercise

Let $S\subset {\mathbb{R}}^2$ be the region bounded by the four curves $x-2y+8=0,$ $x+3y+5=0,$ $x=-2$ and $x=4$. Sketch $S$ and find its centroid.

Exercise

Find the center of mass of trapezoidal object with corners $(x,y)=(2,0)$, $(2,4)$, $(-2,-2)$ and $(-2,0)$ and density $\rho (x,y)=x^2+y+2$.

Exercise

Use polar coordinates to evaluate

$${\int }_0^1\left[{\int }_0^x\sqrt{x^2+y^2}dy\right]dx.$$

Hint: $\int {\sec }^{3}xdx=\frac{1}{2}(\sec x\tan x+\ln |\sec x+\tan x|)+C.$

Exercise

In this question we will evaluate the integral

$$I={\iiint }_{V}10x+8y+6zdxdydz$$

where the integral is over the half ellipsoid

$$V=\{(x,y,z):\frac{x^2}{4}+y^2+z^2\le 1,x\ge 0\}\subset {\mathbb{R}}^3.$$

We choose a change of coordinates $x=r\cos \theta$, $y=\frac{1}{2}r\sin \theta$, $z=z$.

1. Find the Jacobian of this transformation.
2. Find the set of $(r,\theta ,z)$ which corresponds to $V$.
3. Using these, evaluate the integral $I$.

Exercise

The set $V=\{(x,y,z):x^2+y^2\le 4^2,0\le z\le 4-\sqrt{x^2+y^2}\}$ is a cone of height $4$ with base in the $xy$-plane.

The set $W=\{(x,y,z):(x-2{)}^2+y^2\le 4\}$ is a cylinder.

Find the volume of the set $D\subset {\mathbb{R}}^3$ which is the subset of the cone $V$ which is contained within the cylinder $W$.

Hint: Use polar coordinates $x=r\cos \theta$, $y=r\sin \theta$ to write the volume of $D$ as

$${\iint }_{T}r(a-r)drd\theta .$$

for some $a\in \mathbb{R}$, $T\subset {\mathbb{R}}^2$.

Exercise

Let $V$ be the solid bounded above by the sphere $x^2+y^2+z^2=4$ and below by the cone $z=\sqrt{x^2+y^2}$.

Find the volume of $V$.