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  • Solving Integrals for e^-ax^2: (i), (ii) (iii) - Physics Forums
    The discussion revolves around evaluating integrals of the form \ (\int_ {0}^ {\infty} e^ {-ax^2} x^n dx\) for \ (n = 2, 3, 4\), given the known integral \ (\int_ {0}^ {\infty} e^ {-ax^2} dx = \frac {\sqrt {\pi}} {2\sqrt {a}}\) Participants explore methods such as differentiation with respect to the parameter \ (a\) and integration by parts Participants discuss using differentiation with respect to \ (a\) for integrals (i) and (iii), while considering integration by parts for (ii) There
  • Understanding Integration of 1 (x^2 + a^2) and the Role of the Tan . . .
    2) use the substitution , which gives , the indefinite integral you already solved (up to a constant 1 a) Just to be clear, all my integrals here are indefinite integrals
  • What is the relationship between the integral and the area of half a . . .
    The discussion revolves around the relationship between integrals and the area of geometric shapes, specifically focusing on the area of half a circle and an ellipse Participants explore the implications of certain integrals and their geometric interpretations Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation Participants discuss the integral \ (\int_ {-\sqrt {r^2-x^2}}^ {\sqrt {r^2-x^2}} \sqrt {r^2-x^2-y^2}dy\) and its connection to the area of half a
  • How can I solve integrals of the form x^n e^ (-x^2) by hand?
    The discussion revolves around integrals of the form \ (\int x^n e^ {-x^2} dx\), particularly in the context of quantum mechanics The original poster expresses difficulty in solving these integrals by hand and references a standard integral from their textbook They note discrepancies between their results and those from computational tools Exploratory, Conceptual clarification, Mathematical reasoning Participants discuss integration techniques such as integration by parts and
  • Surface Integrals: Flux of F across S - Physics Forums
    The discussion focuses on evaluating the surface integral ∫∫S F · dS for the vector field F (x, y, z) = xy i + yz j + zx k over the surface S defined by the paraboloid z = 3 - x² - y², constrained within the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 The user attempted to convert the integral into polar coordinates but faced confusion regarding the correct application of the surface integral formula and the limits of integration The correct approach involves directly using Cartesian
  • Calculating an integral norm in L2 - Physics Forums
    The operator norm of the integral operator \ ( T \) defined on \ ( H = L^2 (0,1) \) by \ ( Tf (s) = \int_0^1 (5s^2t^2 + 2) (f (t))dt \) is calculated using the relation \ ( ||T|| \leq \sqrt {\frac {50} {6}} \) By taking \ ( f = 1 \), the equality \ ( ||T|| = \frac {\sqrt {65}} {3} \) is established The operator \ ( T \) is self-adjoint due to its symmetric kernel \ ( k (s,t) = 5s^2t^2 + 2 \), and its norm can be determined by finding an orthonormal basis for the two-dimensional subspace





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