Chapter 5

Suppose that you have a positive function and you approximate the area under it using Riemann sums with midpoint rectangles. Explain why, if the function is linear, you will always get the exact area, no matter how many (or few) rectangles you use. [Hint: Make a sketch.]

Evaluate \(\int_{1}^{1} \frac{x^{43} e^{-17 x}+219 \sqrt[3]{x^{2}}}{\ln \sqrt[29]{6 x^{3}-x^{-11}}-\pi^{3}} d x .\) [Hint: No work necessary.

Find a formula for \(\int_{a} c d x .\) [Hint: No calculation necessary-just think of a graph.]

Finding \(\int_{-1}^{1} \frac{1}{x^{2}} d x\) by the Fundamental Theorem of Integral Calculus gives an answer of -2 , as you should check. However, shouldn't the area under a positive function be positive? Explain.

How will \(\int_{a}^{b} f(x) d x\) and \(\int_{b}^{a} f(x) d x\) differ? [Hint: Assume that they can be evaluated by the Fundamental Theorem of Integral Calculus, and think how they will differ at the "evaluate and subtract" step.

ECONOMICS: Pareto's Law The economist Vilfredo Pareto \((1848-1923)\) estimated that the number of people who have an income between \(A\) and \(B\) dollars \((A

BIOMEDICAL: Poiseuille's Law According to Poiseuille's law, the speed of blood in a blood vessel is given by \(V=\frac{p}{4 L v}\left(R^{2}-r^{2}\right)\) where \(R\) is the radius of the blood vessel, \(r\) is the distance of the blood from the center of the blood vessel, and \(p, L,\) and \(v\) are constants determined by the pressure and viscosity of the blood and the length of the vessel. The total blood flow is then given by $$ \left(\begin{array}{c} \text { Total } \\ \text { blood flow } \end{array}\right)=\int_{0}^{R} 2 \pi \frac{p}{4 L v}\left(R^{2}-r^{2}\right) r d r $$ Find the total blood flow by finding this integral \((p, L, v,\) and \(R\) are constants)

BUSINESS: Capital Value of an Asset The capital value of an asset (such as an oil well) that produces a continuous stream of income is the sum of the present value of all future earnings from the asset. Therefore, the capital value of an asset that produces income at the rate of \(r(t)\) dollars per year (at a continuous interest rate \(i\) ) is $$ \left(\begin{array}{c} \text { Capital } \\ \text { value } \end{array}\right)=\int_{0}^{T} r(t) e^{-i t} d t $$ where \(T\) is the expected life (in years) of the asset. Use the formula in the preceding instructions to find the capital value (at interest rate \(i=0.06\) ) of an oil well that produces income at the constant rate of \(r(t)=240,000\) dollars per year for 10 years.

BUSINESS: Capital Value of an Asset The capital value of an asset (such as an oil well) that produces a continuous stream of income is the sum of the present value of all future earnings from the asset. Therefore, the capital value of an asset that produces income at the rate of \(r(t)\) dollars per year (at a continuous interest rate \(i\) ) is $$ \left(\begin{array}{c} \text { Capital } \\ \text { value } \end{array}\right)=\int_{0}^{T} r(t) e^{-i t} d t $$ where \(T\) is the expected life (in years) of the asset. Use the formula in the preceding instructions to find the capital value (at interest rate \(i=0.05)\) of a uranium mine that produces income at the rate of \(r(t)=560,000 t^{1 / 2}\) dollars per year for 20 years.