Double Integral Examples

Here it is... the first post where we(Spring 2011 AMAT 214 class) all collaborate to provide examples for the current course material. You can sign in to blogger with the username: albanymath@yahoo.com, and the password I send in your email(and announced in class) see the last post for how I would prefer you to add your examples. I'll go quickly over in class (and maybe make a video?) showing you how to do it more in depth.

Here are the first examples.


1. Find the volume of the solid under the plane $x+y+z=3$ and above the region in the first quadrant bounded by $y=x^2$ and $y=x^4$. solution

2. Find the volume for the region bounded by the cone $z=\sqrt{2x^2+2y^2}$ and the sphere $$x^2+y^2+z^2=12$$.

Solution:

3. Evaluate by converting the integral to polar coordinates.
$$\int_{-2}^{2}\int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}xy^2~dA$$
Solution:

4. Find the area of the region bounded by the ellipse $16x^2+25y^2=100$ by using
$$A(D)=\iint_{D}dA,$$ with the transformation $x=\frac{u}{4}$ and $y=\frac{v}{5}$. Solution.

5. Evaluate the integral
$$\int_{R}\sin\left(\frac{y-x}{y+x}\right)dA,$$
where $R$ is the triangular region with vertices $(0,0)$, $(1,0)$, and $(0,1)$.
Solution: We'll use the substitution $u=y-x$ and $v=y+x$. This transforms $R$ into a triangular region in the $u-v$ plane with vertices $(0,0)$, $(1,1)$, and $(-1,1)$, which can be described in terms of $u$ and $v$ as follows:
$$-v\leq u\leq v$$
$$-1\leq v\leq 1$$
so we have:
$$\int_{R}\sin\left(\frac{y-x}{y+x}\right)dA=\int_0^1\int_{-v}^{v}\sin\left(\frac{u}{v}\right)\frac{\partial(x,y)}{\partial(u,v)}dudv$$
$$=\int_0^1\int_{-v}^{v}\sin\left(\frac{u}{v}\right)\frac{1}{2}dudv=\frac{1}{2}\int_0^1 -v\cos\left(\frac{u}{v}\right)_{-v}^{v}~dv$$
$$=\int_0^1 \frac{-v}{2}(\cos(1)-\cos(-1))dv=0$$


There you have it. I'll let these go for a week or so and then I will post solutions. In the coming weeks I may scale the webassign back a little to give you time to do this. One last thing, maybe you should "sign up" for the examples in the comments section so no two people are working on the same question.