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Aug 17, 2024 · Consider the function \(f(x,y) = \frac{e^y}{y}\) over the region \(D = \big\{(x,y)\,: 0 \leq x \leq 1, \space x \leq y \leq \sqrt{x}\big\}.\) Notice that the function is nonnegative and continuous at all points on \(D\) except \((0,0)\). Use Fubini’s theorem to evaluate the improper integral.
Aug 17, 2024 · Calculate the volume of a solid of revolution by using the method of cylindrical shells. Compare the different methods for calculating a volume of revolution. In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution.
Aug 17, 2024 · Recognize and use some of the properties of double integrals. Evaluate a double integral over a rectangular region by writing it as an iterated integral. Use a double integral to calculate the area of a region, volume under a surface, or average value of a function over a plane region.
1 day ago · The hypothesis of dark matter has an elaborate history. [18] [19] In the appendices of the book Baltimore lectures on molecular dynamics and the wave theory of light where the main text was based on a series of lectures given in 1884, [18] Lord Kelvin discussed the potential number of stars around the Sun from the observed velocity dispersion of the stars near the Sun, assuming that the Sun ...
Sep 6, 2024 · Picard iterations. Picard's theorem on the existence and uniqueness of solutions of first order differential equations was discussed previously in the introductory section. Let us outline the main ideas in Picard's iteration procedure, starting with the first order differential equation. y = f(x, y), y(x0) = y0.
3 days ago · DSolve finds the general solution for the given ODE. A rule for the function that satisfies the equation is returned. Let’s look at the first order differential equation dy/dx = y(x) d y / d x = y (x) that defined an exponential function as its solution. Indeed, once we enter.
6 days ago · Answer. An intercept equation of a plane is of the form 𝑥 𝑎 + 𝑦 𝑏 + 𝑧 𝑐 = 1. Adding 16 to each side of the given equation, we find 1 6 𝑥 + 2 𝑦 + 8 𝑧 = 1 6. As we want to have 1 on the right-hand side, we divide both sides by 16, which gives 𝑥 1 + 𝑦 8 + 𝑧 2 = 1.
