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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)/04%3A_Appendices/4.01%3A_A%3A_High_School_Material/4.1.01%3A_A.1%3A_Similar_TrianglesTwo triangles T_1,T_2 are similar when (AAA — angle angle angle) The angles of T_1 are the same as the angles of T_2\text{.} (SSS — side side side) The ratios of the side lengths are the s...Two triangles T_1,T_2 are similar when (AAA — angle angle angle) The angles of T_1 are the same as the angles of T_2\text{.} (SSS — side side side) The ratios of the side lengths are the same. That is \begin{align*} \frac{A}{a} &= \frac{B}{b} = \frac{C}{c} \end{align*} (SAS — side angle side) Two sides have lengths in the same ratio and the angle between them is the same. For example \begin{align*} \frac{A}{a} &= \frac{C}{c} \text{ and angle $\beta$ is same} \end{align*}
- https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/01%3A_Integration/1.13%3A_More_Integration_ExamplesRecall that we are using \log x to denote the logarithm of x with base e\text{.} In other courses it is often denoted \ln x\text{.}
- 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
