MAT 321 Calculus 3
Name:
Lab 7: 16.4-16.7
Write detailed solutions to cach of the problems below. Prioritize problem solving and articulate your work clearly, as if you were eqplaining your solution to the reader. Submit your final suhmission as a PDE to the appropriate D2L. Drophox. Unless otherwise communicated, all lahs are due a week from when they are assigned
1. Rectangular Coordinater Rewrite the following integrals using the indicated order of integration. Evaluate the resulting integral.
(a) \( \int_{0}^{1} \int_{-2}^{2} \int_{0}^{\sqrt{4-y^{2}}} d z d y d x \) in the onder of dydzds
(b) \( \int_{1}^{t} \int_{z}^{\mathrm{tz}} \int_{0}^{x^{3}} \frac{\sin \sqrt{y z}}{2^{-1 / 2}} d y d x i z \) to a different order of inlegration Hint: consider the integrand and select an order of integration that allows for simpler integration:
2 Cylindrinal Coordinates
(a) State any formulas needed for cylindrical coordinatex.
(b) Find the volume of the solid in the finst octant bounded by the cylinder \( r=1 \) and the planes \( z-x \) and \( z-0 \).
3. Spherical Coordinate:
(a) State any formulas needed for spherical coordinater
(b) Find the volume of the solad inside the cone \( z=\left(x^{2}+y^{2}\right)^{1 / 2} \) that lies between the planes \( z=1 \) and \( z=2 \)
Figure 1: The cone and planes given in Question 3(b).
4.- Integrals for Mass Calculations:
(a) State the formulas for center of mass in one, two, and three dimensions:
(b) Find the mass and centroid (center of mass) of the thin plate of uniform density bounded by \( \mathrm{g}=\ln x \), the \( x \)-axiv, and \( x=c \)
Hint take advantage of symmetry whenever possible to simplify werk
5. Evaluate the integral
\[
\iint_{R}-x y d A
\]
where \( R \) is the square with vertices \( (0,0),(4,4),(8,0),(4,-4) \) using \( T: x=2 v+2 v, 3= \) \( 2 \mathrm{u}-2 \mathrm{v} \)
(a) Sketuh \( R \) in the \( x y \)-plane.
(b) Sketch the new region \( S \) in the ur-plane using \( T \).
(c) Find the limits of integration for the new integral with respect to \( \mathrm{Ec}, \mathrm{t}= \)
(d) Compute the Jacobian.
(c) Change variables and evaluate the new integral. Hint: Recoll the formula
\[
\iint_{R} f(x, y) d x d y=\iint_{5} J[g(u, v), h(u, v)]|J(u, v)| d u d v,
\]
