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Thread: Using the definite integral

  1. #1
    Jan 2014

    Question Using the definite integral

    The heart pumps blood throughout the body, the arteries are the blood vessels carrying
    blood away from the heart, and the veins return blood to the heart. Close to the heart, arteries
    are very large. The smallest blood vessels are capillaries
    Precise measurements demonstrate that the flux (rate of flow) of fluid through the capillary
    wall is not constant over the length of the capillary. Fluids in and around the capillary are
    subjected to two forces. The hydrostatic pressure, resulting from the heartís pumping, pushes
    fluid out of the capillary into the surrounding tissue. The oncotic pressure drives absorption
    in the other direction. Along the length of
    the capillary, the hydrostatic pressure decreases while the oncotic pressure is approximately
    Using the definite integral-.jpgUsing the definite integral-30-02.gif
    Along a cylindrical capillary of length L = 0.1 cm and radius r = 0.0004 cm, the hydrostatic
    pressure, ph, varies from 35 mm Hg at the artery end to 15 mm Hg at the vein end. (mm
    Hg, millimeters of mercury, is a unit of pressure.) The oncotic pressure, po, is approximately
    23 mm Hg throughout the length of the capillary
    (a) Find a formula for The hydrostatic pressure as a function of x, the distance in centimeters from the artery end of the capillary, assuming that The hydrostatic pressure is a linear function of x.
    (b) Find a formula for p, the net outward pressure, as a function of x.
    (c) The rate of movement, j, of fluid volume per capillary wall area across the capillary wall is
    proportional to the net pressure.We have j = k*p where k, the hydraulic conductivity, has
    Using the definite integral-ab.png
    Check that j has units of volume per time per area.
    (d) Write and evaluate an integral for the net volume flow rate (volume per unit time) through
    the wall of the entire capillary.
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  2. #2
    MHF Contributor
    Mar 2010

    Re: Using the definite integral

    For (a), you are given ph=35 at x=0 and ph=15 at x=0.1, so since the variation is linear, $\displaystyle \text{ph}(x)=35-200x$. The 200 is $\displaystyle \frac{35-15}{0.1}$, which is what it needs to be to get ph=15 at x=0.1.

    In (b), is it correct to just subtract the oncotic pressure? If that's correct, then $\displaystyle p(x)=(35-200x)-23=12-200x$.

    The units of k are the same as j/p, j is in cm/sec and p is in mm Hg, so k is cm/(sec mm Hg).

    For part (d), you are integrating over the length of the capillary, x=0 to x=0.1. Call the volume flow rate Q. The volume flow rate through a small piece of the capillary with length dx is j times p(x) times the area. You get p(x) from part (b) and the area is the area of the side of a cylinder of radius r=0.0004cm and height dx, so it's $\displaystyle 2\pi{r}\,dx$. Putting it together gives $\displaystyle \int_0^{0.1}2\pi{r}jp(x)\,dx$, and since r and j are constants, $\displaystyle 2\pi{r}j\int_0^{0.1}p(x)\,dx$. I assume you can take it from there.

    - Hollywood
    Thanks from gain8823
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