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- https://math.libretexts.org/Courses/Coastline_College/Math_C280%3A_Calculus_III_(Tran)/04%3A_Multiple_Integration/4.07%3A_Calculating_Centers_of_Mass_and_Moments_of_InertiaFind the mass, moments, and the center of mass of the lamina of density \(\rho(x,y) = x + y\) occupying the region \(R\) under the curve \(y = x^2\) in the interval \(0 \leq x \leq 2\) (see the follow...Find the mass, moments, and the center of mass of the lamina of density \(\rho(x,y) = x + y\) occupying the region \(R\) under the curve \(y = x^2\) in the interval \(0 \leq x \leq 2\) (see the following figure). The moment of inertia \(I_x\) about the \(x\)-axis for the region \(R\) is the limit of the sum of moments of inertia of the regions \(R_{ij}\) about the \(x\)-axis.
- https://math.libretexts.org/Courses/Coastline_College/Math_C280%3A_Calculus_III_(Tran)/04%3A_Multiple_Integration/4.08%3A_Change_of_Variables_in_Multiple_Integrals_(Jacobians)\[ \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} \nonumber \\ \dfrac{\partial x}{\partial v} & \dfrac{\partial y}{\partial v} \end{vmatrix} = \left( \frac{\partial x}...\[ \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} \nonumber \\ \dfrac{\partial x}{\partial v} & \dfrac{\partial y}{\partial v} \end{vmatrix} = \left( \frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}\right) = \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \nonumber \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{vmatrix} . \nonumber \]
- https://math.libretexts.org/Courses/Coastline_College/Math_C280%3A_Calculus_III_(Tran)/06%3A_Appendices/6.03%3A_Table_of_Integrals39. \(\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du\) 40. \(\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du\) 70. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u}\,...39. \(\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du\) 40. \(\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du\) 70. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u}\,du=\sqrt{a^2+u^2}−a\ln \left|\frac{a+\sqrt{a^2+u^2}}{u}\right|+C\) 71. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u^2}\,du=−\frac{\sqrt{a^2+u^2}}{u}+\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
- https://math.libretexts.org/Courses/Coastline_College/Math_C280%3A_Calculus_III_(Tran)/05%3A_Vector_Fields_Line_Integrals_and_Vector_Theorems/5.01%3A_Introduction_to_Vector_Field_ChapterVector fields have many applications because they can be used to model real fields such as electromagnetic or gravitational fields. A deep understanding of physics or engineering is impossible without...Vector fields have many applications because they can be used to model real fields such as electromagnetic or gravitational fields. A deep understanding of physics or engineering is impossible without an understanding of vector fields. Furthermore, vector fields have mathematical properties that are worthy of study in their own right. In particular, vector fields can be used to develop several higher-dimensional versions of the Fundamental Theorem of Calculus.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Professor_Dean)/Chapter_5%3A_Integration/5.2_E%3A_Definite_Integral_Intro__Exercises5.2 Exercises - these are good now
- https://math.libretexts.org/Courses/Coastline_College/Math_C185%3A_Calculus_II_(Everett)/04%3A_Techniques_of_Integration/4.03%3A_Trigonometric_IntegralsTrigonometric substitution is an integration technique that allows us to convert algebraic expressions that we may not be able to integrate into expressions involving trigonometric functions, which we...Trigonometric substitution is an integration technique that allows us to convert algebraic expressions that we may not be able to integrate into expressions involving trigonometric functions, which we may be able to integrate using the techniques described in this section. In addition, these types of integrals appear frequently when we study polar, cylindrical, and spherical coordinate systems later. Let’s begin our study with products of sin x and cos x.
- https://math.libretexts.org/Courses/Coastline_College/Math_C185%3A_Calculus_II_(Everett)/03%3A_Applications_of_Integration/3.04%3A_Volumes_of_Revolution-_The_Shell_Method/3.4b%3A_Volumes_of_Revolution-_Cylindrical_Shells_OSIn this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can use this method on the same kinds of solids as the disk method or...In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can use this method on the same kinds of solids as the disk method or the washer method; however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution. With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis of revolution.
- https://math.libretexts.org/Courses/Coastline_College/Math_C185%3A_Calculus_II_(Everett)/04%3A_Techniques_of_Integration/4.10%3A_Chapter_7_Review_ExercisesThese are homework exercises to accompany OpenStax's "Calculus" Textmap.
- https://math.libretexts.org/Courses/Coastline_College/Math_C185%3A_Calculus_II_(Everett)/04%3A_Techniques_of_Integration/4.04%3A_Trigonometric_Substitution/4.4E%3A_Exercises_for_Trigonometric_Substitution\(\displaystyle ∫\frac{\sqrt{x^2−1}}{x^2}\,dx \quad = \quad −\frac{\sqrt{−1+x^2}}{x}+\ln\left|x+\sqrt{−1+x^2}\right|+C\) 49) Find the surface area of the solid generated by revolving the region bounde...\(\displaystyle ∫\frac{\sqrt{x^2−1}}{x^2}\,dx \quad = \quad −\frac{\sqrt{−1+x^2}}{x}+\ln\left|x+\sqrt{−1+x^2}\right|+C\) 49) Find the surface area of the solid generated by revolving the region bounded by the graphs of \(y=x^2,\, y=0,\, x=0\), and \(x=\sqrt{2}\) about the \(x\)-axis. (Round the answer to three decimal places). 50) The region bounded by the graph of \(f(x)=\dfrac{1}{1+x^2}\) and the \(x\)-axis between \(x=0\) and \(x=1\) is revolved about the \(x\)-axis.
- https://math.libretexts.org/Courses/Coastline_College/Math_C280%3A_Calculus_III_(Everett)/03%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/3.04%3A_Partial_Derivatives/3.4E%3A_Partial_Derivatives_(Exercises)These are homework exercises to accompany Chapter 13 of the textbook for MCC Calculus 3
- https://math.libretexts.org/Courses/Coastline_College/Math_C280%3A_Calculus_III_(Everett)/03%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/3.04%3A_Partial_DerivativesFirst, the notation changes, in the sense that we still use a version of Leibniz notation, but the \(d\) in the original notation is replaced with the symbol \(∂\). (This rounded \(“d”\) is usually ca...First, the notation changes, in the sense that we still use a version of Leibniz notation, but the \(d\) in the original notation is replaced with the symbol \(∂\). (This rounded \(“d”\) is usually called “partial,” so \(∂f/∂x\) is spoken as the “partial of \(f\) with respect to \(x\).”) This is the first hint that we are dealing with partial derivatives.
