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MS-A0211 - Differential and integral calculus 2,  period III, 2023 \hfill María C. Quintana \\
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\textbf{Homework for Week 5.} 
To be submitted for marking before the  \textbf{deadline 15.2.2023}. \\See ``Assignments" section in MyCourses for instructions. Please remember that clear and logical mathematical writing is important.



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	\item   Decide whether each of the formulas below makes sense as
	a double integral.  For the formulas that you reject, explain your reasoning
	in one or two sentences; for the legitimate ones, sketch the region of
	integration in the plane and perform one step of the integration (that is, do the inside integral). 
	\begin{align*}
	\text{(a) (1pt)} & \int_0^1 \int_0^{3x} (2x+4y) \, dx \, dy , & \text{(b) (1pt)} & \int_0^1 \int_0^{3x} (2x+4y) \, dy \, dx \\
	\text{(c) (1pt)} & \int_0^{3x} \int_0^1 (2x+4y) \, dx \, dy, 
	& \text{(d) (1pt)} &\int_0^1 \int_{y-1}^{1-y^2} (2x+4y) \, dx \, dy \\
	\text{(e) (1pt)} & \int_0^3 \int_{y-1}^{1-y^2} (2x+4y) \, dx \, dy, 
	& \text{(f) (1pt)} & \int_0^{x^3} \int_{y-1}^{1-y^2} (2x+4y) \, dx \, dy \\
	\end{align*}
	
	\item (2pts) Compute the double integral of the function $$f(x,y) = e^{x^2 + y^2}$$ over the top half of the disk of radius $3$ centered at $(0,0)$.
	
	\item  (2pts) Find the center of mass of a two-dimensional plate that occupies the triangle formed by $x = 0$, $y = x$ and $2x + y = 6$ and has density function $$\rho(x,y) = x^2.$$ Sketch the plate and its center of mass.

	
	

	
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