Help with developing equations for integral applications in physics. I have the question: The soot produced by a garbage incinerator spreads out in a circular pattern. The depth, H(r) , in millimeters, of the soot deposited each month at a distance r kilometers from the incinerator is given by $$H(r)=0.119{e^{-2.1r}}$$ Write a definite integral (with independent variable r) giving the total volume of soot deposited within 5 kilometers of the incinerator each month. I wrote that the integral was $$\int_0^5 \ {2{pi}r}{0.119{e^{-2.1r}}}\,dr.$$ and after evaluating, I said the volume of the soot was 0.0005404606479 km^3. I don't think this answer is correct but I can't seem to see where I'm going wrong. Is the integral set up correctly?

In the integrand, you seem to be multiplying $r$ and $dr$ in kilometers, and then multiplying by $H(r)$, the depth in millimeters. The limits of the integration, and the coefficient in the exponential are also in kilometers. Change the depth to kilometers by dividing $H(r)$ by $10^6$, and the result will be in $km^3$.