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Magazine Problem 11
Find the following area by computing the values of a definite integral: The area bounded by the straight line \(y=2 x+3\), the \(x\) axis, the line \(x=1\), and the line \(x=4\).
Problem 12
Find the following area by computing the values of a definite integral. The area bounded by the parabola \(y=4-x^{2}\) and the \(x\) axis. You will have to find the limits of integration.
Problem 13
Determine whether each of the following improper integrals converges, and if so, determine its value: (a) \(\int_{0}^{\infty} \frac{1}{x^{3}} \mathrm{~d} x\). (b) \(\int_{-\infty}^{0} e^{x} \mathrm{~d} x\).
Problem 14
Determine whether the following improper integrals converge. Evaluate the convergent integrals. (a) \(\int_{1}^{\infty}\left(\frac{1}{x^{2}}\right) d x\). (b) \(\int_{1}^{\pi / 2} \tan (x) \mathrm{d} x\).
Problem 15
Determine whether the following improper integrals converge. Evaluate the convergent integrals (a) \(\int_{0}^{1} \frac{1}{x \ln (x)} \mathrm{d} x\). (b) \(\int_{1}^{\infty}\left(\frac{1}{x}\right) \mathrm{d} x\).
Problem 16
Determine whether the following improper integrals converge. Evaluate the convergent integrals \(\int_{0}^{\pi} \tan (x) \mathrm{d} x\). (a) \(\int_{0}^{\pi / 2} \tan (x) d x\). (b) \(\int_{0}^{1}\left(\frac{1}{x}\right) \mathrm{d} x\).
Problem 17
Determine whether the following improper integrals converge. Evaluate the convergent integrals, (a) \(\int_{0}^{\infty} \sin (x) d x\). (b) \(\int_{-\pi / 2}^{\pi / 2} \tan (x) \mathrm{d} x\).
Problem 18
Approximate the integral $$ \int_{0}^{\infty} e^{-x^{2}} d x $$ using Simpson's rule. You will have to take a finite upper limit, choosing a value large enough so that the error caused by using the wrong limit is negligible. The correct answer is \(\sqrt{\pi} / 2=0.886226926 \cdots\).
Problem 19
Using Simpson's rule, evaluate erf \((2.000)\) : $$ \operatorname{erf}(2)=\frac{2}{\sqrt{\pi}} \int_{0}^{2} e^{-t^{2}} \mathrm{~d} t $$ Compare your answer with the correct value from a more extended table than the table in Appendix G, \(\operatorname{erf}(2.000)=0.995322265 .\)
Problem 20
Find the integral: \(\int \sin [x(x+1)](2 x+1) \mathrm{d} x\).
