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    Integrals

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    Integration

    Integration of Exponential Functions. See attached file for full problem description.

    Integration: Finding Work Done by Gas and Moment of Mass

    4. A quantity of gas with an initial volume of 1 cubic foot and a pressure of 2500 pounds per square foot expands to a volume of 3 cubic feet. Find the work done by the gas for the given volume. Assume that the pressure is inversely proportional to the volume. 6. Find moment of mass M_x, M_y, and center of the mass for the la

    Integrals : Volume of a Hollow Cylinder

    A 6.00 in radius cylindrical rod is 2 ft long. Use a differential to approximate how much nickel (in in^3) is needed to coat the entire rod with the thickness of .12 in. keywords: integrals, integration, integrate, integrated, integrating, double, triple, multiple

    Evaluating a Double Integral

    Evaluate the double integral integral from y=0 to y=3 of integral from x=0 tp 1-y of (x+y)dxdy int ( int (x+y), x=0..1-y), y=0..3)

    Polar Coordinates : Circles and Integrals

    There are three problems in one question. a) in polar coordinates, write equations for the line x=1 and the circle of radius 2 centered at the origin b) write an integral in polar coordinates representing the area of the region to the right of x=1 and inside the circle c) evaluate the integral

    Integrals: Regions and Curves

    Regions and Curves. See attached file for full problem description. Sample problems 1. ∫(e-2x)dx 2. Find area under curve, f(x)=(1/x²) on [1,2] 3. ∫18 5x(2/3)dx 4. ∫19 (√x+(1/√x))dx 5. ∫0√3 x(x²+1)(3/2)¬dx 6. ∫¬35 (3/(x-2))dx 7. Find the area of the region bounded by f(x)=ex, g(x)=(1/2

    Finance/Accounting Problems

    10. Stock Values. Integrated Potato Chips paid a $1 per share dividend yesterday. You expect the dividend to grow steadily at a rate of 4 percent per year. 1. What is the expected dividend in each of the next 3 years? 2. If the discount rate for the stock is 12 percent, at what price will the stock sell? 3. What is the exp

    Surface Integrals

    Find the surface integral (double integral over S) E dot dS, where S is the cylinder, x^2 + y^2 = 4, z is greater than or equal to 2 and less than or equal to 5, and the vector field F is F(x, y, z) = (0, 0, z^2)

    Vector Fields and Surface Integrals

    Find the flux of the vector field F(x, y, z) = (y, 0, z2) out of the unit sphere S. In other words, find the surface integral ∫∫S (y, 0, z2) * dS, where the sphere S is oriented by the outward normal. Let S be the cylinder x2 + y2 = 1, 0 ≤ z ≤ 6. Find ∫∫S (x4 + 2x2y2 + y4)2 dS.

    Conic Surfaces and Surface Integrals

    Let S be the conic surface z = 3 sqrt (x^2 + y^2), where z is greater than or equal to 0 and less than or equal to 3. Find (double integral over S) z dS.

    Areas of Bounded Regions

    Find the area of the region bounded by: (using integrals) a)F (x) = 6x- x^2 and g(x)= x^2 - 2x b) y = x^2 - 4x and y = x-4 keywords: integration, integrates, integrals, integrating, double, triple, multiple

    Change in Revenue Using Integrals

    The marginal revenue for a certain product is given by dR/dx = 25-2x. Find the change in revenue when sales increase from 7 to 10 units. keywords: integration, integrates, integrals, integrating, double, triple, multiple

    Integral Functions

    Find y = f (x) if a) dy/dx = xe^(〖-x〗^2 ) and f(0) = 2 b) f"(x) = 1/e^x + 2 , f'(0) =3, f (0) = 1 keywords: integration, integrates, integrals, integrating, double, triple, multiple

    Double Integral Evaluated

    Evaluate the double integral: / / l l (3x -2y)dA; R is the region enclosed by the circle l l x^(2) + y^(2) = 1 . / /

    Calculus: Evaluating an Integral

    Let C be the curve represented by the equations x = 2 t , y = 3 t^(2). Evaluate the integral (0 <= t <= 1) / l (x - y)ds . l / C

    Lp Spaces Positive Measurements

    Suppose m is a positive measure on X, m(X) < inf f is an element of L^(inf), ||f||_inf<inf and a_n= int (|f|^n)dm for n = 1,2,... where the integral is evaluated over the set X Show that lim as n->inf of (a_(n+1))/(a_n) is equal to ||f||_inf

    Path and Line Integrals

    Find I = &#8747;c y2 dx + y dy, where c: x = cost, y = sint, 0 &#8804; t &#8804; pi. Evaluate the integral I = &#8747;c y dx + x dy, where the path c is given by x = t3 + 1, y = 1/(t3 + 1) and 0 &#8804; t &#8804; 1. keywords: integration, integrates, integrals, integrating, double, triple, multiple

    Cantor Sets

    1. Show that the Cantor function c: [0, 1] → [0, 1] is continuous. To do this, I know I need to use the fact that c is monotone, but I'm having difficulty from there. 2. Compute ∫c, where c is considered to be an element of L+(R). (let c(x) =0 for x not in [0, 1]) Here, c is the Cantor Function and L+(R) consists

    Center of Mass and Length of Path

    1) A plate is bounded by the curves y = 0, y = x^2 + 1, x = -1 and x = +1. It has density d(x,y) = x^2. Find its center of mass (x bar, y bar) 2) Find the length of the path c'. x = 1, y = t, z = t^2/2, t is greater than or equal to 0 and less than or equal to 1.

    Contours and the Cauchy Integral Formula

    Let C be the boundary of the square of side length 4, centered at the origin, with sides parallel to the coordinate axes, and traversed counterclockwise. Evaluate each of the attached integrals.