Let SS be the plane 5x+3y-3z=155x+3y-3z=15 in octant V with a downward facing Normal, and F→=\vec{F}=.a) Evaluate ∫S∫(∇×F→)∙N→ dS\int_S\int{(\nabla\times\vec{F})\bullet\vec{N}\ dS}b) Find ∫CF→∙ dr→\int_C{\vec{F}\bullet\ d\vec{r}} where CC is the boundary curve of SS.Hint: it iS possible to answer This part (b) withOut showing any worK as long as you statE why/what theorem(s) you are uSing

Let QQ be the solid bounded by the paraboloid z=x2+y2z=x^2+y^2 and z=16z=16 with SS as its boundary surface oriented outward as usual. And let F→=\vec{F}=.a) Set up and simplify, with bounds, but do not evaluate, the integrals ∫S∫F→∙N→ dS\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\int\int\limits_Q\int{\nabla\bullet\vec{F}\ dV}

Type your answers in as best you can. Avoid decimals: leave your answers with square roots and/or fractions if applicable. There is a math editor if you click the ⊕\oplus button. Don’t forget to upload your work through the provided link in blackboard or via email, even if you show it here to your recording, because that’s how I will be grading your exams.

Use Green’s Theorem to rewrite, but not evaluate, ∫Cxy2 dx+5x2y dy\int_C{xy^2\ dx + 5x^2y\ dy}, where CC is the boundary around the region given by y=0y=0, x=ex=e, and y=ln(x)y=\ln(x). Include bounds for your integral.

Using polar coordinates, evaluate ∫R∫(x2+y2) dA\int_R\int{\sqrt{(x^2+y^2)}\ dA} where RR is the washer 1≤x2+y2≤91\leq x^2+y^2 \leq 9 in quadrants II, III, and IV.

a) Parametrize, including bounds, the cylinder 4×2+z2=164x^2+z^2=16 with -4≤y≤6-4\leq y \leq 6.b) Parametrize, including bounds, z2+x2-y2=1z^2+x^2-y^2=1 with 0≤y≤30\leq y \leq 3.

Find the surface area of $$\vec{r}(r,\theta)=$$ with 0≤θ≤2π0\leq\theta\leq 2\pi and 0≤r≤60\leq r\leq 6.Hint: ∫∫1 dS=∫∫N→ dA\int\int{1\ dS}=\int\int{\left\|\vec{N}\right\|\ dA}, where N→=r→r×r→θ\vec{N}=\vec{r}_r \times \vec{r}_{\theta}

Module 11 – ADH A patient has a condition that prevents their collecting ducts from responding to antidiuretic hormone (ADH). As a result, the permeability of the collecting duct to water does not change. Under normal conditions, how does ADH affect water permeability in the collecting duct? (4 points) If the collecting duct is always impermeable to water, what effect would this have on urine volume and concentration? (2 points) What might happen if the collecting duct were always permeable to water, regardless of the body’s hydration status? (2 points)

How do influenza and HIV avoid immune system control, and how does this affect long-term immunity?

Module 11 – Loop of Henle If a defect prevents solute transport out of the ascending limb of the Loop of Henle: What is the normal role of the ascending limb in creating the vertical medullary osmotic gradient? (4 points) How would a failure to transport solutes here affect the osmolarity of the medulla? (3 points) How would this change in the medulla impact water reabsorption in the descending limb of the loop of Henle? (4 points) What effect might this have on the overall ability of the kidney to concentrate urine? (3 points)