MATH241 Lecture 32: The Indefinite Integral, Net Change Theorem, and Substitution Rule

8. · 1 points SSTCaic6 5.1.028 My Notes Ask Your Find the indefinite integral. (Remember to use absolute values where appropriate. Use C for the constant of integration,) dx Additional Materials Book 0.5/1 points I The marginal revenue of a certain commodity is RO) =-3x2 + 4x + 32, where SSTCalc6 5.1 My Notes k Your is the level of production (in thousands). Assume R(0) = 0. (a) Find the demand function p(x) px)-12x32 (b) Find the level of production that results in maximum revenue. What is the market price per unit at this level of production? Additional Materials @Book 10. SSTCalc6 5.1.044 My Notes Ask Your A particle travels along thex-axis in such a way that its acceleration at time tis a(e)- Vi*2, If it starts at the origin with an initial velocity of 3 (that is, s(o) o and v(o) -3), determine its position and velocity when t 2. (Round your answers to two decimal places.) s(2)- v(2)- Additional Materials eBook points SSTCalcs 5.1.054. My Notos Ask Your Find the area under the curve defined by the given equetion, above the x-axis, and over the given interval. y- e-xover [0, 6]

Use a linear approximation to estimate the quantity to four decimal places Find the total area of the region between the curve and the x-axis. 2) Find the area of the shaded region 3) 3) y- 2 y - 2sin(rx) 4) y-2x2+x 6 y-x2.4 a. Find the limit. + X 5) Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed 6) f(x) 2 between x . 4 and x-8 using the midpoint sum with four rectangles of equal 6) width.

Solve the initial value problem Graph the integrand and use geometry to evaluate the integral. Compute the definite integral as the limit of Riemann sums 9) Use the substitution formula to evaluate the integral L/2 10)_ 10) 0 (2 5 sin Determine the indefinite integral. Check your work by differentiation. 13 0 6 90 12) sec θ . cos e 1b the conclusion of the Mean Value Theorem tor Find the value or values of e that satisty the equationD- the function and interval. 13 12.9 13) Find the average value of the function over the given interval. 14) y-x2. 5x+6:(0.8 14) Find the pointis) at which the given function equals its average value on the given interval. 15) Evaluate the integral. 16)ル咔+4)dt 16) π7-sin 10x dx 17) 10

Euler defined the function and proved G(n) = n! = 1 middot 2 middot 3 middot middot middot n for all positive integers n. Redo Euler's work by first calculating G(0) = 1 and then use integration by parts to show G(n) = n G(n - 1). That says G(n) = n! by recursion. Using standard methods to reduce to single integrals, evaluate the following double integral in terms of G(m) and G(n): Calculate the Jacobian of the two-variable substitution What is the region S in the ut-plane that maps onto [0, infinity) times [0, infinity) in the xy-plane under the map (u, v) (x, y) = (tu, t(1 - u))? Hint: simplify x + y. Using the substitution in parts (c) and (d) and the change-of-variables theorem, show that Euler used the G(n) integral to invent the definition of n! for non-integers n. Use the result of parts (b) and (e) to find the value of (1/2)!. You may need to evaluate the integral The easiest way to do that is to see that is a very simple curve, whose area is known.