Chapter 6: Problem 16
In Exercises \(11-20,\) solve the initial value problem explicitly. $$\frac{d y}{d x}=5 \sec ^{2} x-\frac{3}{2} \sqrt{x}\( and \)y=7\( when \)x=0$$
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Finding Area Find the area of the region enclosed by the \(y\) -axis and the curves \(y=x^{2}\) and \(y=\left(x^{2}+x+1\right) e^{-x}\)
Group Activity Making Connections Suppose that $$\int f(x) d x=F(x)+C$$ (a) Explain how you can use the derivative of \(F(x)+C\) to confirm the integration is correct. (b) Explain how you can use a slope field of \(f\) and the graph of \(y=F(x)\) to support your evaluation of the integral. (c) Explain how you can use the graphs of \(y_{1}=F(x)\) and \(y_{2}=\int_{0}^{x} f(t) d t\) to support your evaluation of the integral. (d) Explain how you can use a table of values for \(y_{1}-y_{2}\) \(y_{1}\) and \(y_{2}\) defined as in part (c), to support your evaluation of the integral. (e) Explain how you can use graphs of \(f\) and \(\mathrm{NDER}\) of \(F(x)\) to support your evaluation of the integral. (f) Illustrate parts (a)- (e) for \(f(x)=\frac{x}{\sqrt{x^{2}+1}}\) .
Integral Tables Antiderivatives of various generic functions can be found as formulas in integral tables. See if you can derive the formulas that would appear in an integral table for the fol- lowing functions. (Here, \(a\) is an arbitrary constant.) See below. (a) \(\int \frac{d x}{a^{2}+x^{2}} \quad\) (b) \(\int \frac{d x}{a^{2}-x^{2}} \quad\) (c) \(\int \frac{d x}{(a+x)^{2}}\)
In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{0}^{7} \frac{d x}{x+2}$$
Average Value A retarding force, symbolized by the dashpot in the figure, slows the motion of the weighted spring so that the mass's position at time \(t\) is $$y=2 e^{-t} \cos t, \quad t \geq 0$$ Find the average value of \(y\) over the interval \(0 \leq t \leq 2 \pi\)
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