Integral Calculus Problems 8/26/2019 rd

2-Variable Function with a Maximum z = f(x,y) 8/26/2019 rd

Sequence Problem Solving ?? 49 36 18 8 2 5 11 17 23 ? 3 3 5 4 4 3 5 ? 77 7*7 = 49 Skip every other prime 31 1 2 2 4 2 4 2 4 ?

2-Variable Function with both Maxima and Minima z = f(x,y) 8/26/2019 rd

2-Variable Function with a Saddle Point z = f(x,y) 8/26/2019 rd

Natural Logarithms With x/h = n tending to infinity yielding e 8/26/2019 rd

Integration Integral Calculus Stochastic (Probability) Models Differential Equations Dynamic Models 8/26/2019 rd

Integration If F(x) is a function whose derivative F’(x) = f(x), then F(x) is called the integral of f(x) For example, F(x) = x3 is an integral of f(x) = 3x2 Note also that G(x) = x3 + 5 and H(x) = x3 – 6 are also integrals of f(x) 8/26/2019 rd

Indefinite Integral The indefinite integral of f(x), denoted by where C is an arbitrary constant is the most general integral of f(x) The indefinite integral of f(x) = 3x2 is 8/26/2019 rd

…or use a table of integrals A Strategy Guess and Test the integrand …or use a table of integrals 8/26/2019 rd

Methods Of Integration Integrating Power Functions Fundamental Arithmetic Integration Rules Basic Integration Formulas Tables of Integrals Non-Integrability Partial Fractions Integration by Parts xe-xdx = -xe-x + e-x dx let u = x; dv = e-x dx du = 1; v = -e-x 8/26/2019 rd

Integration by Parts d(uv) = udv + vdu udv = uv - vdu Show that xnexdx = xnex - nxn-1exdx + C let u = xn; dv = exdx then du = nxn-1dx; v = ex + C Thus, xnexdx = xnex - nxn-1ex dx + C 8/26/2019 rd

The top five 8/26/2019 rd

Basic Rules of Integration 8/26/2019 rd

The top four and the basic rules in action… 8/26/2019 rd

Initial Conditions The rate at which annual income (y) changes with respect to years of education (x) is given by where y = 28,720 when x = 9. Find y. 8/26/2019 rd

Integrating au , a > 0 8/26/2019 rd

Helpful Methods 8/26/2019 rd

Use some algebra 8/26/2019 rd

Adjusting for “du” – method of substitution 8/26/2019 rd

More du’s 8/26/2019 rd

Integrate x(x - 1)1/2 8/26/2019 rd

Partial Fractions 8/26/2019 rd

Integration by Parts derived from the product rule for derivatives 8/26/2019 rd

Another one? 8/26/2019 rd

Integration by Tables A favorite integration formula of engineering students is: 8/26/2019 rd

Check it out! 8/26/2019 rd

Another Table Problem 8/26/2019 rd

An Engineer’s Favorite Table 8/26/2019 rd

The Definite Integral Areas under the curve 8/26/2019 rd

Definite Integral Given a function f(x) that is continuous on the interval [a,b] we divide the interval into n subintervals of equal width, x, and from each interval choose a point, xi*.  Then the definite integral of f(x) from a to b is 8/26/2019 rd

Area under the curve f(x) x x 8/26/2019 rd

The Fundamental Theorem of Calculus Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be a function such that                    for all x in [a, b] then                                  . 8/26/2019 rd

Fundamental Theorem 8/26/2019 rd

Evaluating a definite integral 8/26/2019 rd

Changing Limits 8/26/2019 rd

The Area under a curve The area under the curve of a probability density function over its entire domain is always equal to one. Verify that the following function is a probability density function: 8/26/2019 rd

Area between Curves Find the area bounded by y = 4 – 4x2 and y = x2 - 1 y1 = 4 – 4x2 y1 - y2 = 5 – 5x2 (-1, 0) (1, 0) y2 = x2 - 1 8/26/2019 rd

Area Find area bounded by y2 – x = 0 and y – x + 6 = 0. Curves intersection at y2 – y – 6 = 0; (y-3)(y+2) (x – 6)2 - x = 0 => (x – 9)(x – 4) = 0 (9,3) (4, -2) (+ 32/3 54/3 -81/2 54 -16/3 8 -24) -->125/6 8/26/2019 rd

Rectilinear Motion A particle moves right from the origin on the x-axis with acceleration a = 5 – 2t and v0 = 0. How far does it go? a = 5 – 2t => v = 5t – t2 + v0 => s = 5t2/2 – t3/3 v = 0 when = 5t – t2 = 0 or when t = 5 s(5) = 125/2 -125/3 = 125/6 ft 8/26/2019 rd

Y = x; x2; x3 8/26/2019 rd

Improper Integrals 8/26/2019 rd

Example – an Improper Integral 8/26/2019 rd

Let’s do another one… 8/26/2019 rd

The Engineers Little Table of Improper Definite Integrals 8/26/2019 rd

Some Applications Taking it to the limit… 8/26/2019 rd

The Crime Rate The total number of crimes is increasing at the rate of 8t + 10 where t = months from the start of the year. How many crimes will be committed during the last 6 months of the year? 8/26/2019 rd

Learning Curves Cumulative Cost hours to produce ith unit cumulative direct labor hrs to produce x units average unit hours to produce x units 8/26/2019 rd

Learning Curves Approximate Cumulative Cost 8/26/2019 rd

Learning Curves - example Production of the first 10 F-222’s, the Air Force’s new steam driven fighter, resulted in a 71 percent learning curve in dollar cost where the first aircraft cost $18 million. What will be cost of the second lot of 10 aircraft? (sim-lc 18e6 20 71) (sim-lc 18e6 20 71) 8/26/2019 rd

The Average of a Function The average or mean value of a function y = f(x) over the interval [a,b] is given by: Find the average of the function y = x2 over the interval [1,3]: 8/26/2019 rd

Average profit An oil company’s profit in dollars for the qth million gallons sold is given by P = P(q) = 369q – 2.1q2 – 400 If the company sells 100 million gallons this year, what is the average profit per gallon sold? 8/26/2019 rd

An Inventory Problem Demand for an item is constant over time at the rate of 720 per year. Whenever the on-hand inventory reaches zero, a shipment of 60 units is received. The inventory holding cost is based upon the average on-hand inventory. Let y = 60 – 720t be the on-hand inventory as a function of time where t is in years. It takes 60/720 = 1/12 yr to go from an inventory of 60 to 0. 60 t 8/26/2019 rd

Annuities A is the present value of a continuous income stream Let A = the present value of a continuous annuity at an annual rate r (compounded continuously) for T years if a payment at time t is at the rate of f(t) per year. Then A is the present value of a continuous income stream 8/26/2019 rd

Annuity Example Determine the present value of a continuous annuity at an annual rate of 8% for 10 years if the payment at time t is at the rate of 1000t dollars per year. 8/26/2019 rd

Annuities Let S = the accumulated amount of a continuous annuity at an annual rate r (compounded continuously) for T years if a payment at time t is at the rate of f(t) per year. Then 8/26/2019 rd

Back to the example 8/26/2019 rd

More of that example Recall continuous compounding 8/26/2019 rd

Iterated Integrals Evaluate 8/26/2019 rd

Double Integral Evaluate the integral over R where R is the triangle formed by y = x, y = 0, x = 1. 8/26/2019 rd

f(x, y) = 2 for 0 < x < y < 1 Find 8/26/2019 rd

Area between Regions Find the area bounded by y = x3 - 4x and y = 3x + 6. -2 -1 3 Curves intersect at (cubic 1 0 -7 -6)  (3 -1 -2) 8/26/2019 rd

Quadratic Equation Derivation ax2 + bx + c = 0 x2 + (b/a)x + (c/a) = 0 Complete the square x2 + (b/a)x + (c/a) + (b2/4a2) = (b2/4a2) (x + b/2a)2 = -c/a + b2/4a2 x = -b/2a + (- c/a + b2/4a2)1/2 = - b  (b2 – 4ac)1/2 2a 8/26/2019 rd

Partial Fractions Solve for A and B by the cover-up method and integrate to get 3 ln(x - 2) + 2 ln(x + 4) + C or ln(x - 2)3(x + 4)2 + C 8/26/2019 rd

Sketch the Plane 2x + 3y + 4 = 12 z 3 4 y 6 x 8/26/2019 rd

Tangent Find the equation of the tangent to the curve y = x ln x – x where x = 1. y' = 1 + ln x – 1 = ln x or ln 1 = 0 y = -1 at x = 1 with slope 0. (y +1) = 0(x – 1) => y = -1 8/26/2019 rd

Predator/Prey # prey attack y = K(1 - ex), K and  constants dy/dx = Kex and ex = -y/K +1 => dy/dx = K(-y/K + 1) = (K – y) 8/26/2019 rd

Average Value of a Function over [a, b]. Find the average value of f(x) = 2 – 3x2 over [-1, 2] 8/26/2019 rd