Blog

Treating an auto-immune disease such as Ulcerative Colitis or Lupus by infecting the diseased person with a tapeworm or a pinworm would be in agreement with which of the following ideas?

Come up with your own non-constant conservative vector field F→\style{font-size:35px}{\vec{F}}. Show that is it conservative. Then, find the work done by F→\style{font-size:35px}{\vec{F}} over the curve starting at (5,5)\style{font-size:35px}{(5,5)}, looping around the arrow on the x-\style{font-size:35px}{x-}axis, visiting Neptune, traveling to another universe, then coming back and ending up back at (5,5)\style{font-size:35px}{(5,5)}.Hint: work can be represented by ∫CF→∙ dr→\style{font-size:35px}{\int_C{\vec{F}\bullet\ d\vec{r}}}.

Let Q\style{font-size:35px}{Q} be the solid bounded by the paraboloid z=x2+y2\style{font-size:35px}{z=x^2+y^2} and plane z=16\style{font-size:35px}{z=16} with S\style{font-size:35px}{S} as its boundary surface oriented outward as usual. And let F→=\style{font-size:35px}{\vec{F}=}.a) Set up and simplify, with bounds, but do not evaluate, the integrals ∫S∫F→∙N→ dS\style{font-size:35px}{\int_S\int{\vec{F}\bullet\vec{N}\ dS}}Hint: the flat top of the solid is its own function and requires its own integralb) Use the divergence theorem to set up and evaluate the integral ∫∫Q∫∇∙F→ dV\style{font-size:35px}{\int\int\limits_Q\int{\nabla\bullet\vec{F}\ dV}}

Set up an integral in spherical coordinates that represents the volume of the sphere x2+y2+z2=25x^2+y^2+z^2=25 in octant VI. Include bounds for your integral, but no need to evaluate.

Set up an integral in spherical coordinates that represents the volume of the sphere x2+y2+z2=49x^2+y^2+z^2=49 in octant VII. Include bounds for your integral, but no need to evaluate.

Evaluate ∫1e ∫ln(y)1e(x2)y dxdy\int_1^{e}\ \int_{\ln(y)}^{1}{\frac{e^{(x^2)}}{y}\ dxdy} by switching the order of integration.

Using polar coordinates, evaluate ∫R∫ln(x2+y2) dA\int_R\int{\ln{(x^2+y^2)}\ dA} where RR is the washer 4≤x2+y2≤94\leq x^2+y^2 \leq 9 in quadrants I, II, and III.

Find the surface area of y2+z2+2x=0y^2+z^2+2x=0 with -2≤x≤0-2\leq x\leq 0.

head muscles.jpg  

Screenshot (71).png