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- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.07%3A_G%3A_Solutions_to_Exercises
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/02%3A_Applications_of_Integration/2.02%3A_AveragesAnother frequent application of integration is computing averages and other statistical quantities. We will not spend too much time on this topic — that is best left to a proper course in statistics —...Another frequent application of integration is computing averages and other statistical quantities. We will not spend too much time on this topic — that is best left to a proper course in statistics — however, we will demonstrate the application of integration to the problem of computing averages.
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/01%3A_Integration/1.04%3A_SubstitutionIn the previous section we explored the fundamental theorem of calculus and the link it provides between definite integrals and antiderivatives. Indeed, integrals with simple integrands are usually ev...In the previous section we explored the fundamental theorem of calculus and the link it provides between definite integrals and antiderivatives. Indeed, integrals with simple integrands are usually evaluated via this link.
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A%3A_High_School_Material/4.1.13%3A_A.13%3A_LogarithmsIn the following, x and y are arbitrary real numbers that are strictly bigger than 0, and p and q are arbitrary constants that are strictly bigger than one. \(q^{\log_q x}=x, \qquad \l...In the following, x and y are arbitrary real numbers that are strictly bigger than 0, and p and q are arbitrary constants that are strictly bigger than one. qlogqx=x,logq(qx)=x logqx=logpxlogpq logq1=0,logqq=1 logq(xy)=logqx+logqy logq(xy)=logqx−logqy lim
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A%3A_High_School_Material/4.1.03%3A_A.3%3A_Trigonometry_%E2%80%94_Definitions\begin{array}{rlcrl} \sin\theta &= \dfrac{\text{opposite}}{\text{hypotenuse}} & \qquad & \csc \theta &= \dfrac{1}{\sin\theta} \\ \cos\theta &= \dfrac{\text{adjacent}}{\text{hypotenuse}} & \qquad & \se...\begin{array}{rlcrl} \sin\theta &= \dfrac{\text{opposite}}{\text{hypotenuse}} & \qquad & \csc \theta &= \dfrac{1}{\sin\theta} \\ \cos\theta &= \dfrac{\text{adjacent}}{\text{hypotenuse}} & \qquad & \sec \theta &= \dfrac{1}{\cos\theta} \\ \tan\theta &= \dfrac{\text{opposite}}{\text{adjacent}} & \qquad & \cot \theta &= \dfrac{1}{\tan\theta} \end{array} \nonumber \]
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A%3A_High_School_Material/4.1.12%3A_A.12%3A_PowersIn the following, x and y are arbitrary real numbers, and q is an arbitrary constant that is strictly bigger than zero. q^{x+y}=q^xq^y\text{,} q^{x-y}=\frac{q^x}{q^y} \(q^{-x}=\fra...In the following, x and y are arbitrary real numbers, and q is an arbitrary constant that is strictly bigger than zero. q^{x+y}=q^xq^y\text{,} q^{x-y}=\frac{q^x}{q^y} q^{-x}=\frac{1}{q^x} \lim\limits_{x\rightarrow\infty}q^x=\infty\text{,} \lim\limits_{x\rightarrow-\infty}q^x=0 if q \gt 1 \lim\limits_{x\rightarrow\infty}q^x=0\text{,} \lim\limits_{x\rightarrow-\infty}q^x=\infty if 0 \lt q \lt 1
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A%3A_High_School_Material/4.1.04%3A_A.4%3A_Radians%2C_Arcs_and_SectorsFor a circle of radius r and angle of \theta radians: Arc length L(\theta) = r \theta\text{.} Area of sector A(\theta) = \frac{\theta}{2} r^2\text{.}
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/02%3A_Applications_of_Integration/2.03%3A_Centre_of_Mass_and_TorqueIf you support a body at its center of mass (in a uniform gravitational field) it balances perfectly. That's the definition of the center of mass of the body.
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/01%3A_Integration/1.12%3A_Improper_IntegralsTo this point we have only considered nicely behaved integrals \int_a^b f(x)\, d{x}\text{.} Though the algebra involved in some of our examples was quite difficult, all the integrals had
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Sequence_and_series/3.04%3A_Absolute_and_Conditional_ConvergenceWe have now seen examples of series that converge and of series that diverge. But we haven't really discussed how robust the convergence of series is — that is, can we tweak the coefficients in some w...We have now seen examples of series that converge and of series that diverge. But we haven't really discussed how robust the convergence of series is — that is, can we tweak the coefficients in some way while leaving the convergence unchanged.
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.03%3A_C%3A_More_About_Numerical_Integration/4.3.02%3A_C.2%3A_Romberg_Integrationwhich says that B(h) is an approximation to \cA whose error is of order k+1\text{,} one better 2 That is, the error decays as h^{k+1} as opposed to h^k — so, as h decreases, it...which says that B(h) is an approximation to \cA whose error is of order k+1\text{,} one better 2 That is, the error decays as h^{k+1} as opposed to h^k — so, as h decreases, it gets smaller faster. We are next going to consider a family of algorithms that extend this idea to use small step sizes in the part of the domain of integration where it is hard to get good accuracy and large step sizes in the part of the domain of integration where it is easy to get good accuracy.
