Sketch the curve in polar coordinates and find the area of the inner loop.

Sketch the curve in polar coordinates given by r = 2-4 sin theta Find the area of the inner loop. Find the area of the inner loop for the general case: i.e. r = b - a sin feta (0 less than b less than a )

Use a transformation to evaluate the double integral

Use a transformation to evaluate the double integral of f(x,y) given by f(x,y)=cos(2x-y)sin(x+2y) over the square region with vertices at O(0,0) P(1,-2) Q(3,-1) & R(2,1) (My notes from class-uses substitution, change of variables)

Transformation

Please see the attached file for full problem description. --- Use a transformation to evaluate the double integral of f(x,y) given by f(x,y)=cos(2x+y)sin(x-2y). over the square region with vertices at (0,0) P(1,-2) Q(3,-1) & R(2,1) (My notes from class-uses substitution, change of variables) Solution. Letting ...continues

Practical Calculus Problem for the Design of Water Guttering

Please see the attached file for the fully formatted problems. MAX FLOW A company is constructing guttering to carry water. The cross section of the guttering is below: Each side is the same length and the angle between each side and the hosizontal are equal. That is, the cross section is symmetrical about the vertical ...continues

Partial Derivatives

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First Principle in evaluating derivatives

(A) Find and simplify the difference quotient for G(X)=1/x^2. HINT: After finding the difference quotient, simplify by using an LCD to combine the fractions. (B) Using the answer above, find the value of the difference quotient at x=1 with an h=.1 C) Sketch a graph of G(x). Mark the point(1,G(1)) on the graph. Sketch a ...continues

Parametric Equation

Professional golfer Nancy Lopez hits a golf ball with a force to produce an intinial velocity of 175 feet per second at an angle of 35 degrees above the horizontial. She estimates the distance to the hole to be 225 yards. A) Write the position of the ball as a pair of parametric equations. (I know that it is x=175tcos(35)and ...continues

Polar Coordinates : Solving Derivatives and Circuit Problem

Please see the attached file for full problem description. (a) By making the substitution y = z/x^4, or otherwise, reduce the equation dy/dx +4y/x =sinx/x2 to an equation in which the variables are separable. Solve the equation if y = 0 when x = pi/2 (b) In a circuit di/dt=K(E-Ri) and i=0 when t=0. Find i in ...continues

Multivariate Calculus

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Derivatives using product rule, quotient rule, or chain rule.

Find the derivative of each expression, using the product rule, quotient rule, or chain rule. 1. P= e^(2x)/x 2. B= square root of sin * square root of x 3. Find dy/dx using implicit differentiation. (3xy + 1)^5 = x^2