Definite Integrals Questions and Answers

2 Find the volume of the solid region bounded by the graphs of the equations z 9 x y 3x y 3 x 0 y 0 z 0

xample 1 1 Sex dx Use the CSR Asw with n 8 to approximate 113 0 0315

2 Given f x 4x 5 and g x x 5 write an equation for g f x g f x 4 x 5 5 g f x 2 x a a 3 Let f x 10x 1 Write an expression for f x 2 f x C C 10 8 b d 2 20 b d

Use the piecewise function to evaluate the following H a f 4 a c f 3 e f 0 b f 1 d f 2

Evaluate the line integral 0 3 0 2 03 2 01 02 2 3yzds where Cis the curve with parametric equations 2 t y sint z cost 0

5 4 3 2 4 3 2 1 y 2 3 4 5 12 3 4 5 Find the inequality represented by the graph I

Evaluate the line integral PF dr by evaluating the surface integral in Stokes Theorem with an appropriate choice of S Assume that C has a counterclockwise orientation when C F 5y z x C is the circle x y 22 in the plane z 0 Rewrite the given line integral as an area integral over the appropriate region of the xy plane SS 5 da dA R C dr Evaluate the integral Er Type an exact answer COLOS

Evaluate the surface integral f x y z ds using an explicit representation of the surface S f x y z 4xy S is the plane z 1 x y in the first octant Set up the surface integral for the given function f x y z over the given surface S Integrate with respect to x first Use increasing limits of integra 00 S S O dx dy 00 SS f x y z ds S Type exact answers

Select the equivalent expression 1 3 p 12

Find the area of the surface of the half cylinder r 0 z r 3 0 0sx 0 zs 7 using a parametric description of the surface Set up the integral for the surface area using the parameterization u 0 and v z SSO 00 Type an exact answers using a as needed du dv

and the area of the following surface using the given explicit description of the surface z 3 x y2 for 0 sz 108 The paraboloid z Set up the surface integral for the given function over the given surface S as a double integral over a region R in the xy plane SS as SSO S R Tyne an exact answer using as needed dA

Use the Binomial Theorem to expand and simplify the expression y 9 4

11 Find t e 3x dx a c diverges 0 b d

18 The region between the curve y x 0 x 4 and the x axis is revolved about the x axis to generate a solid Find its volume a c 2TT 47 b d 8T 67

9 5 points Use the limit process to find the area of the region bounded by the graph of the function and the x axis over the given interval y 5 r 2 1 Sketch the region

Evaluate the definite integral f2r x 1 dx

4 Find the area of the region enclosed by r 2 cos 0 a b 7 4 c 4 9 2 FORT HUBERT S d 5T

Let R be the region in the xy plane bounded between y 4 x and y 1 x Set up an integral that is equal to the volume of the solid obtained by revolving R about the line x 5 using a the cylindrical shell method and b the washer method Do not evaluate the integral s

As blood moves through a vein or an artery its velocity v is greatest along the central axis and decreases as the function of r is called the law of laminar flow For an artery with radius 0 5 cm the relationship between v in cm s and r in cm is given by the function v r 18 500 0 25 2 Osrs0 5 0 5 cm a Find v 0 1 and v 0 4 v 0 1 4440 v 0 4 16650 cm s x cm s b What do your answers to part a tell you about the flow of blood in this artery The blood flows much faster 0 1 cm from the center than 0 1 cm from the edge The blood flows much slower 0 1 cm from the center than 0 1 cm from the edge The blood flows with the same velocity 0 1 cm from the center as it does 0 1 cm from the edge Nothing can be determined about blood flow from part a c Make a table of values of v r for r 0 0 1 0 2 0 3 0 4 0 5 r in cm V r in cm s 0 4625 0 1 4440 0 2 3885 0 3 2960 0 4 1665 0 5 0

27 Find a formula for the nth term of the arithmetic sequence 6 2 2 6 10 OA 6 4n B 6 2n O C 10 2n D 10 4n

The formula A 21 20 0409 a What was the population of the state in 2000 b When will the population of the state reach 29 8 million models the population of a US state A in millions t years after 200

4 Find P a da P x dx for k 2 3 4 5 and 6

6 10 pts Use Stokes Theorem to evaluate the integral f F dr where x y z ayi z2j 3y2 k and C is the trace of the cylinder z y2 9 in the plane 2r 8 oriented counterclockwise as viewed from above

c Consider the solid in the first octant cut off from the square cylinder with sides x 0 x 1 y 0 and y 1 by the plane x y z 4 Find the volume 3 ways i A da dz dy integration 15 points ii A dz dy dx integration 15 points

Use the surface integral in Stokes Theorem to calculate the flux of the curl of the field F across the surface S in the direction away from the origin F 4yi 5 5x j z 2 k S r 0 2 sin cos 0 i 2 sin osin 0 j 2 cos 4 k 0 r 2 0 0 2

Find the domain of f and write your answer in interval notation f x 2x 9 x 12x 32

What is the third term in the binomial expansion of 3x y 4 O 54x O 18x y3 O 18x 6 O 54x y6

The region R in the first quadrant is bounded by the graph of y = tan(x), the x-axis, and the vertical line x = 1. What is the volume of the solid formed by revolving R around the vertical line x = 1? 0.3751 0.7012 1.1783 2.203

For the polynomial function below, answer parts (a) through (d). (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of |x|. f(x)=8(x² +9) (x²+8) ² GLITOR (a) Find any real zeros of f. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. The real zero(s) of f is/are (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. There are no real zeros.

1. Evaluate the iterated integral: 3 √9-x^2 ∫ ∫ (x² + y²)dy dx 0 -√9-x^2

Evaluate the definite integral S 8 sin(x)dz

Let f(x, y) = x sin(x² + y) +x+y for z, y € (-00, 00). Which of the following statement is true? (a) 9² (b) f(x, y)dy = -x cos(z² + a) + x cos(x²) + ax + for a > 0. 2 f(x,y) lim (2,3)+(0,0) x(x² + y) = 1. (c)f(x, y) = 2x² cos(x² + y) + sin(x² + y) + 1. (d) The tagent planc to the surface z = f(x, y) at the point (0,0,0) is z = 2x + y. (e) None of the above statements is truc.

If ["* f(z)dz = [ f(z)dz + ["* f(z)dz – - 2 what are the bounds of integration for the first integral? 0= and b= e-,f(t)dt,

If 5 is the hemisphere a² + y² +2²= 25 with z>0 then the surface integal f(x-2y)ds is__

How many significant figures are in the number 21.2? 21.2 has [?] significant figures.

Solve the following system of equations (use any method): 2x-5y=11 -3x+4y=-13 O (-2,-3) O (3.-1) O(-3.1)

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x = (y - 6)², x = 25; about y = 1

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. x = 4y², y ≥ 0, x = 4; about y = 2

Apply Laplace transform method improper integral from 0 to infinity ∫e^-3t sint cost dt

What is the amplitude of the following function? y = 0.918 cos x

(0.00/1.00 point) Use a system of linear equations with two variables and two equations to solve. A concert manager counted 350 ticket receipts the day after a concert. The price for a student ticket was $12.50, and the price for an adult ticket was $16.00. The register confirms that $5,075 was taken in. How many student tickets and adult tickets were sold? Let a represent the total number of student tickets sold, and let y represent the total number of adult tickets sold. Give you answer as an ordered pair (x, y).

Solve the equation by applying the square root property. If there is more than one solution, enter them in order from smallest to largest. Be sure to write a set as the answer. Do not enter any spaces in the answer. (x + 5)² = 9 The solution set is A/

Solve the equation, then select all true statements about the equation. 4(x-7)= 4x - 28 a) The solution set is the empty set. b) The equation is a conditional equation. c) The solution set is {0}. d) The solution set is the set of all real numbers. e) The equation is a contradiction. ntion is an identity.

9. Use the washer method to the find the volume of the solid of revolution formed by revolving the region between y = r² and y = √ around the z-axis. Set the two functions equal to each other to determine the limits of integration. Draw a sketch to determine the upper and lower functions.

Determine the volume of the solid generated by rotating function f(z) = [4,10]. 100 -z² about the x-axis on -

Givenc (x is number of items) Demand function: (z) Supply function: a(z)=2./2 Find the equilibrium quantity: 2200 Find the consumers surplus at the equilibrium quantity: 9

= Sketch the region enclosed by y = 7x and y 6x2. Find the area of the region.

18 A company's marginal cost function is where x is the number of units. Find the total cost of the first 49 units (of increasing production from x=0 to x=49) Total cost: S

10. Use the washer method to the find the volume of the solid of revolution formed by revolving the region between y = 1 and y = r around the r-axis. Set the two functions. equal to each other to determine the limits of integration. Draw a sketch to determine the upper and lower functions.

A curve has equation y = f(x). (a) Write an expression for the slope of the secant line through the points P(7, f(7)) and Q(x, f(x)). f(x) - x f(7)-7 O O X-7 f(x)-f(7) Of(x)-f(7) x-7 O f(7) -7 f(x) X (b) Write an expression for the slope of the tangent line at P. f(x)-f(7) X-7 O lim x-0 X-7 x-7 f(x)-f(7) Olim Olim f(x) = f(7) x-7 f(x) - x x-0 (7)-7 Olim