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Integrals

Riemann Integration, Partitions, Upper and Lower Sums

1. Suppose f: [a,b] &#61614;&#61522; is a function such that f(x)=0 for every x &#61646;(a,b]. a) Let &#61541; > 0. Choose n &#61646; &#61518; such that a + 1/n < b and |f(a)|/n <&#61541;. Let P ={a, a+1/n, b} &#61646; &#61520;([a,b]). Compute &#61525;(f,P) - &#61516;(f,P) and show that is less than &#61541;. b) Prove

Computing areas and volumes using multiple integrals.

(1) Find the volume of the solid bounded by the paraboloid x2 + y2 = 2z, the plane z = 0 and the cylinder x2 + y2 = 9. (2) Find the volume of the region in the first octant bounded by x + 2y + 3z = 6. (3) Find the area of the solid that is bounded by the cylinders x2+z2 = r2 and y2+z2 = r2. (4) Find the volume enclosed by t

Applications of the Change of Variables Theorem

(1) Find ... (x + y)2 dx dy...where R is the square with vertices (±1, 0) and (0,±1), (2) Let R now be the triangular region in the xy plane with vertices (1, 0), (2, 1), (3, 0). Find.... (3) Change the integral .... from rectangular to polar coordinates. See the attached file.

Indefinite Integral Functions

A) Find the indefinite integrals of the following functions. Please see attached questions, please can you show your working to help me understand. thanks

Revenue Function and Definite Integral of Revenue Function

Revenue at day D = (200 + 10D - 100P)*P D refers to the day, with Monday being 1, Tuesday 2, etc up to Friday with a value of 5. The assignment is as follows: 1. If you are charging $1 per cup, what is your revenue for each of the five days? What is your total revenue for the week? 2. What is the indefinite integral

Stokes Theorem, Curl and Positively Oriented Hemisphere

7) Use Stoke's Theorem to evaluate curl F*dS S is the hemisphere oriented in the direction of the positive x-axis. 8) Use Stokes Theorem to evaluate C is the boundary of the part of the plane 2x + y + 2z = 2 in the first octant. 9) Suppose that f(x,y,z)= , where g is a function of one variable such that g(2) = -

Green's Theorem, Positively Oriented Curve, Ellipse

Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. ∫c xdx + ydy. C consists of the line segments from (0,1) to (0,0)...and the parabola y = 1 -x^2.... Use Green's theorem to evaluate the line intgral along the positively oriented curve. ∫c sin y dx + x cos y dy

Integration Techniques and Applications

1. Find the indefinite integrals for the following functions: a. f(X) = 10000 b. f(X) = 20X c. f(X) = 1- X2 d. f(X) = 5X + X-1 e. f(X) = 12- 2X f. f(X) = X3 + X4 g. f(X) = 200X - X2 + X100 2. Find the definite integral for the following functions: a. f(X) = 67 over the interval [0,1] b.

Force and Opposing Force : Find maximum speed attained

A particle of mass 10kg, moving in a straight line, starts at rest from a point A under the action of a force that decreases uniformly from 20N to zero in 20 secs. It then travels with a constant speed for a further 20s, and finally moves under the action of an opposing force of 40N until it comes to rest at B. Find the maximum

Description of Definite Integral as the Limit of a Sum

Calculus Integral Calculus(III) Definite Integral as the Limit of a Sum Method of Summation Definite Integral It is an explanation for finding the integral by the method of summation or by evaluating the integral as limit of a sum (part 3). Find by the method of summation the value of: &#8747; sin xdx, where the lo

Definite Integral as the Limit of a sum BrainMass Expert explains

Calculus Integral Calculus(I) Definite Integral as the Limit of a sum Method of Summation Definite Integral It is an explanation for finding the integral by using the method of summation(part I). Find by the method of summation the value of: (a) ∫ e^( - x)dx, where the lower limit is a and the upper limit is b

Implicit Differentiation,Graph of Inequality and Extremes

Find the indefinite integral and check by differentiation &#8747;x^3+2 dx Solve the inequality and sketch the graph of the solution on the real number line x- 5 > or = 7 Find the extreme of the function of the closed interval f(x) = 5- 2x^2, [0,3] Find dy/dx using implicit differentation for y^3 -x^2 + xy=3