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This document discusses formulas for calculating area and arc length using polar coordinates. The three essential formulas are: 1) area of a polar region using infinitesimal wedges, 2) area between two polar graphs by subtracting the inner area from the outer area, and 3) arc length of a polar curve considering a small section of arc length between two theta values. Examples are given such as finding the area of a rose petal or a cardioid. The document ends with an application problem calculating the number of WikkiStix needed to make a heart shape using the equation r = 1 - sinθ.
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Polar area
1) The area enclosed by a polar curve can be found by integrating the area of sectors swept out by the radius as the angle varies over the interval.
2) The area of each sector is 1/2*r^2*Δθ. Integrating this expression from α to β gives the total area.
3) Special care must be taken when curves intersect or complete multiple cycles over the interval to break the problem into multiple integrals if needed. Graphing and inspecting the curves is important to set up the integrals correctly.
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The document discusses topological dimension and fractal dimension. It provides examples of objects with different dimensions, such as a line being 1-dimensional, a plane being 2-dimensional, and the Cantor set having a fractal dimension of 0.6309. Fractals are self-similar objects that can have non-integer dimensions calculated using the logarithmic formula D = log(N)/log(1/r), where N is the number of parts and r is the scaling ratio. The Koch snowflake is used to illustrate this, with N=4, r=1/3, giving a fractal dimension of 1.26.
Triple integrals in spherical coordinates
1. The document discusses triple integrals in spherical coordinates. A point in xyz space is characterized by spherical coordinates ρ, θ, and φ which are related to x, y, and z by equations that switch between rectangular and spherical coordinates.
2. A typical triple integral in spherical coordinates has the form of an integral with bounds of ρ from 0 to some value R, θ from 0 to 2π, and φ from 0 to π, integrating a function f(ρ, θ, φ) multiplied by the Jacobian ρ2sin(φ).
3. Switching between rectangular and spherical coordinates involves equations that express x, y, z in terms of ρ, θ, φ or vice versa
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11 4 Areas Of Regular Polygons
The document discusses calculating the areas of regular polygons. It states that the area of a regular polygon is equal to half the product of the apothem and perimeter. The apothem is the distance from the center of the polygon to the midpoint of one side, and can be calculated using trigonometric functions like tangent and cosine based on the central angle of the polygon. Formulas are provided for calculating the areas of regular pentagons and general regular n-gons.
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2) The area of each sector is 1/2*r^2*Δθ. Integrating this expression from α to β gives the total area.
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Module#9 notes
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Module#9 notes
The document discusses the standard equation forms of a hyperbola depending on whether it has a horizontal or vertical transverse axis. For a hyperbola with a horizontal transverse axis, the standard equation is (x-h)^2/a^2 - (y-k)^2/b^2 = 1. For a hyperbola with a vertical transverse axis, the standard equation is (y-k)^2/a^2 - (x-h)^2/b^2 = 1. The document also lists key properties of hyperbolas including the center, vertices, foci, and asymptotes.
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Power point presentation PIYUSH BHANDARI
3-D shapes have length, width, and height. A cube has 6 faces, 12 edges, and 8 vertices. The surface area of a cube can be found by calculating the area of each face, which is the length squared, and multiplying by 6 since there are 6 faces. The surface area of other 3-D shapes like cuboids and cylinders can be found by calculating the total area of all faces, such as using the formula for a cuboid of 2 times the length times width plus 2 times the height times width plus 2 times the length times height. The surface area of a cylinder is 2 times pi times the radius times the height plus 2 times pi times the radius squared.
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3) Examples are provided to demonstrate applying the formulas step-by-step to calculate surface areas and volumes.
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Лекция 2 Сортировки, поиск и порядковые статистики
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Hand book of Howard Anton calculus exercises 8th edition
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The document discusses calculating the area of a region bounded by a polar curve. It explains that the area can be approximated using Riemann sums of sectors of circles, where the radius of each sector is given by the polar curve. The exact area is given by the limit of these Riemann sums as the number of sectors approaches infinity. The formula for the area of a region bounded by a polar curve r=f(θ) from θ=α to θ=β is ∫_α^β 1/2 r^2 dθ. Several examples are worked out applying this formula to find the area of regions defined by polar equations.
Differential calculus
1) The document discusses concepts in differential calculus including the product rule, quotient rule, derivatives of trigonometric functions, higher-order derivatives, and polar curves.
2) Polar curves are represented using polar coordinates (r, θ) where r is the distance from the origin and θ is the angle. Conversions between polar and Cartesian coordinates are covered.
3) Methods for calculating the area inside a polar graph, length of a polar curve, and surface area obtained by rotating a polar curve about the x-axis are presented using integral calculus.
Trapezoidal rule
The Trapezoidal rule approximates the area under a curve by dividing it into trapezoids and calculating their individual areas. It works by taking the ordinates at evenly spaced intervals along the x-axis and using the formula: Area = (1/2) * (first ordinate + last ordinate + 2 * sum of middle ordinates) * width. This provides an estimate of the definite integral. The more trapezoids used, the more accurate the estimate. The estimate will be an overestimate if the gradient is increasing and an underestimate if decreasing over the interval.
Kernel Methods on Manifolds
This document discusses manifolds and kernels on manifolds. It defines manifolds and different types of manifolds such as topological manifolds, differentiable manifolds, and Riemannian manifolds. It then discusses Hilbert spaces, kernels, and reproducing kernel Hilbert spaces. It explains that defining kernels on manifolds allows applying kernel methods to nonlinear manifolds. It discusses challenges in defining positive definite kernels on manifolds using geodesic distance and provides conditions for when the Gaussian RBF kernel is positive definite on a manifold. It also covers applications to pedestrian detection and visual object categorization using kernels on the manifold of symmetric positive definite matrices.
5.13.1 Area of Circles, Sectors, and Quads
Develop and use formulas to find the areas of circles and sectors.
Develop and use formulas to find the areas of special quadrilaterals
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This document discusses methods for computing the area and volume of surveyed land. It describes several rules that can be used to calculate area based on the shape and accuracy needs, including mid-ordinate rule, average ordinate rule, trapezoidal rule, and Simpson's one-third rule. The trapezoidal and prismoidal rules are presented for calculating volume from cross-sectional areas. The most accurate methods involve subdividing irregular boundaries into geometric shapes or using a planimeter for highly irregular boundaries.
Computation of Area and volume
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AndersSchreiber_ThesisFinal
- The author derives the Schwarzschild metric in D spatial dimensions and one time dimension to investigate the effects of a fractal deviation from three dimensions.
- Using the metric, the author considers phenomena like perihelion precession, bending of light, and gravitational redshift to set bounds on the parameter δ that quantifies the deviation from three dimensions.
- The best upper bound found is |δ| < 5.0 × 10-9, from measurements of perihelion precession in binary pulsar systems. This is consistent with zero deviation from three dimensions.
Single Variable Calculus Assignment Help
I am Duncan V. I am a Single Variable Calculus Assignment Solver at mathhomeworksolver.com. I hold a Master's in Mathematics, from Manchester, United Kingdom. I have been helping students with their assignments for the past 5 years. I solved assignments related to Single Variable Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Single Variable Calculus Assignments.
Polar co
The polar coordinate system represents each point on a plane using a distance (radial coordinate r) from a fixed point (pole) and an angle (angular coordinate φ) from a fixed direction (polar axis). While polar coordinates are not inherently unique, conventions define r as non-negative and φ within a range of 360° or 2π radians to ensure a unique representation for each point. Polar equations define curves as r as a function of φ, and different forms of symmetry can be deduced from the polar equation. Common curves like circles, lines, roses, spirals, and lemniscates have relatively simple polar equations compared to their Cartesian forms.
Lar calc10 ch04_sec2
The document discusses methods for approximating and calculating the area of plane regions. It introduces sigma notation for writing sums and explores using rectangles to approximate the area of a region bounded by a graph and the x-axis. Upper and lower sums are defined by using maximum and minimum heights of a function over subintervals, and the area is defined as the limit of the Riemann sums as the subintervals approach zero width. Examples demonstrate finding upper and lower sums and calculating the exact area of a region using limits.
Differential Calculus
The document covers several topics related to polar curves and higher-order derivatives:
- It discusses how to find higher-order derivatives by taking successive derivatives of a position function. For example, to find acceleration, take the second derivative of a position function.
- It provides an overview of the polar coordinate system, including how to plot points using polar coordinates and how to convert between polar and rectangular coordinates.
- It shows how to calculate the area enclosed by a polar curve by integrating r over the angle limits, and provides an example of calculating this area.
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- 1. area and arc length of polar coordinates. 5.24.10 monicaagueand patricia liu.
- 2. formulas. Wewon’t be distributing a formula sheet, butthesethree formulas are essentiallyallyouwillneedto know to do anyproblemregarding polar areaor arc length. Ifyouwouldliketomakeyourownteeny-tiny formula sheet, yourthreeessential formulas are: areaof a polar region. areabetween 2 polar graphs. lengthof a polar arc.
- 3. area of a polar region. Uses infinitelysmallwedges, eachofwhichhaveanarea, a radiusdefined by r=f(θ), and a widthdefined by Δθ. ~or~
- 6. areabetweentwo polar graphs. Findtheareaoftheregioninsidethecardioidandoutsidethecircle: Cardioid: Circle: Set equationsequaltooneanother in ordertofindpointsofintersection: Solve:
- 7. area betweentwo polar graphs. Subtract the area of the “inside” curve from “outside” curve and solve:
- 8. findtheareaoftheregioncommontothecircleandthecardioid. Whensolvingforareainsidetwo curves, addthetwoareasoneachinterval. Thiswillresult in one-halfofthe total areaenclosed by the curves. Circle: Cardioid:
- 9. find the area of theregion common to the circle and thecardioid.
- 10. lengthof apolar arc. Considering a polar arc r = f(θ); α≤θ≤β whichalsoequals….
- 12. Findthelengthofthe arc fromθ= 0 toθ= 2π forthecardioid
- 13. application. ItisValentine’s Day and Billy theMath Geekisdeterminedtomaketheperfect Valentinecardfortheprettiest girl in class, Cindy. He plansonconstructing a heartonthefrontofthecardusingWikkiStix. Likethetruemathgeekthat he is, Billy plansonusingtheequationr = 1 - sinθtomakehisheart. EachWikki Stick is 5 inches long. How manyWikkiStixwill Billy needtoearnCindy’saffection?
- 14. workscited. Anton, Howard, and Albert Herr. Calculus with Analytic Geometry. New York: Wiley, 1995. Print. Anton, Howard, Stephen Davis, and IrlBivens. Calculus. New York: John Wiley & Sons, 2002. Print. "Arc Length/Polar Area." Peterson's AP Calculus AB/BC. Lawrenceville, NJ: Peterson's, 2005. Print. Areas and Lengths Using Polar Coordinates. Dir. Selwyn Hollis. Perf. Selwyn Hollis. Areas and Lengths Using Polar Coordinates. University of Houston, 2005. Web. 23 May 2010. Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. "Area and Arc Length of Polar Coordinates." Calculus of a Single Variable. Boston: Houghton Mifflin, 1998. Print. Stewart, James, Harvey Keynes, and Dan Clegg. "Areas and Lengths in Polar Coordinates." Single Variable Calculus. Belmont, CA: Thomson Brooks/Cole, 2003. Print.