Definite Integrals Questions and Answers

0 0 1 0 1 0 State what the shaded area represents A Total cost in dollars B Rate of change of area C Total area in square feet D Rate of change of price 1 point s possible Cost dollars per square foot Area square feet

and the area of the shaded region x x x 20x g x 0 6 5 4 3 2 50 40 30 20 10 1 20 30 40 50 1 2 3 4 5 6 C Next questio A OB OC OD 2521 12 1625 12 224 3 243 4

Next que 0 A company determines that its marginal revenue per day is given by R t 140 e R 0 0 where R t the revenue in dollars on the tth day The company s marginal cost per day is given by C t 130 0 3t C 0 0 where C t the cost in dollars on the th day Find the total profit from t 0 to t 6 the first 6 days T Round to the nearest cent Note P T R T C T R 1 C 1 dt 0 0 1 0 1 A 55 554 63 B 55 705 43 C 55 570 83 D 55 565 43

0 1 0 0 1 0 1 0 1 0 1 Find the area under the graph of the function over the interval given 1 y 1 4 x OA 2 O B 1 2 O c 4 1 O D 4

The period is Enter NONE if not periodic help numbers Click

0 Use geometry to evaluate the definite integral 0 10 0 1 0 1 0 1 8 0 8x dx OA 256 B 68 OC 32 OD 512

O 1 0 1 0 1 0 1 Evaluate the definite integral and interpret the result 16 S x 3 dx 0 4 3 2 1 1 3 10 12 14 16 E Next question OA 16 the shaded area above the x axis minus the shaded area 3 below the x axis is equal to 16 3 M B 16 O C 16 the total shaded area the shaded area above the x axis minus the shaded area 3 below the x axis is equal to OD 38 the total shaded area 3 16 3

A graph of f x is shown below 10 10 4 0 10 3 2 1 2 5 4 1 f x 1 2 5 Suppose g x f 2x 3 5 Using the graph above and the transformations ap

2 2 Sketch the volume bounded by the surfaces z 1 y 2z 2 in the first quadrant and set up the bounds for an integral over this region in three ways where this volume is projected onto the ry plane the yz plane and the arz plane When you project to the ry plane setting the integral up is a bit different Why is it different and how did you handle that difference

DISTANCE Inches 10 9 8 7 3 2 1 0 0 1 2 3 4 6 7 8 DISTANCE Inches 5 The green upper triangle has an area of 6 inches 9 10 The purple lower triangle has an area of 6 inches 6 inches 7 inches 7 inches 12 inches 12 inches Place the orange triangle square symbols directly next to the green triangle so that the two triangles together make a rectangle The total a this rectangle is which is the area of the green triangle

ar UE as ur tte Find the area of the shaded region f x 20x x x g x 0 The area is Type an integer or a simplified fraction

Find the area of the shaded region f x x 8x 24x g x 12x 45 The area is Type an integer or a simplified fraction

Find the area of the shaded region f x 6x x x g x 0 The area is Type an integer or a simplified fraction

Find the area of the shaded region f x x 8x 21x g x 6x 36 The area is Type an integer or a simplified fraction Next question 1 30 60 19 10

ar JE S ur te Find the area of the region bounded by the graphs of the given equations y x y x x 0 x 1 Set up the integral s that will give the area of the region Select the correct choice below and fill in any answer box es to complete the choice O A O B Odx dx dx 0 5 N C

S Find the area of the region bounded by the graphs of the given equations y 4 y x x 0 The area of the region is Simplify your answer

A company determines that its marginal revenue per day is given by R t where R t is the total accumulated revenue in dollars on the tth day The company s marginal cost per day is given by C t where C t is the total accumulated cost in dollars on the fth day R t 130 et R 0 0 C t 130 0 5t C 0 0 a Find the total profit P T from t 0 to t 10 the first 10 days T So R 1 C 1 dt Round to the nearest cent as needed P T R T C T The total profit is RAMB

8 Determine whether both of the following improper integrals converge or diverge If it converges specify the value it converges to 4 points 2 0 12 1 1 2 dx 1 6 dx 2 2 x Hint The integrand in b has a discontinuity at x 2

Determine the following f4e 4e 0 4x 0 4x dx dx Type an exact answer

T 9 Use the comparison theorem for improper integrals to prove that 2 points cos x x7 dx converges

4 Evaluate 8x 12x2 7x 2 2x 2 dx

Suppose f is an even function with Evaluate each integral 2 6 1 a 1 x f x 2x 1 dx b f 0 f x dx 3 3 2 cos x f 2 1 sin x 1 sin x dx

Does the series converges conditionally converges absolutely O diverges 1 k7 5 k 1 Does the series k 1 1 k 37 5 O converges absolutely O converges conditionally O diverges converge absolutely converge conditionally or diverge converge absolutely converge conditionally or diverge

Find the points on the ellipse 4x 2y 1 where f x y xy has its extreme values The points are Type an ordered pair Use a comma to separate answers as needed Type an exact answer using radicals as neec

Find the radius of convergence for X 3x n 1 n

7n e for convergence you can use the Integral Test This is also a geometric series To test the series so we could also investigate convergence using other methods Find the value of of the 7x dx What does this value tell you about the convergence of the series X n 1 e O the series might converge or diverge we need more information O the series definitely converges O the series definitely diverges 7n

Evaluate the integral 3 S T s 5 2 S 2 S S ds 3 23 S 2 S S ds

ind the total area of the shaded regions Ay 18 16 4 12 10 8 6 4 2 0 2 2 3 y ex e 5 X Q

Question Find the values of the six trig functions for the triangle below First we need to find the length of the hypotenuse Select Now select the correct values below sin 0 Select Select cos Select 50 Select Select Select Select Select

Estimate the value of the following integral using n 4 in each case Round answers to six decimal places 1 cos x dx n 4 a Trapezoidal Rule

A chain lying on the ground Is 10 m long and its mass is 80 kg One end of the chain is raised to a height of 6 m a Draw before and after pictures of this scenario That is draw one picture of the chain lying on the ground Then draw a second picture of the chain with one end raised to a height of 6 m Choose where dy a tiny portion of the chain should be and clearly label dy on both pictures Also clearly label distance y in your after picture Note a distance such as y has a starting point and an ending point show both D

Evaluate the indefinite integral Show full work justify fully dt cos t 1 9 tan t D Only use methods taught in lecture

Calculate the definite integral 2 2 f8e 9x 0 gx dx

1 Find the area between y 2x and y x from x 0 to x 3 The area is

Calculate the definite integral 2 2 S 7 0 7e5x 7 5x dx C

Find the volume V of the solid below the paraboloid z 4 x y and above the following region R r 0 1 r 2 0 0 2 C

Graph f locate all points of discontinuity and discuss the behavior of f at these points 3 x if x 1 1 x if x 1 f x Choose the correct graph of f x below OA B Locate all points of discontinuity Select the correct choice below and if necessary fill in the answer box to complete your choice OA The point s of discontinuity is are x Use a comma to separate answers as needed OB There are no points of discontinuity O C OD Ay

Find the area of the region bounded above by y 4 and below by y x 5 O O w w w woo 16 O 3 O 36 59 60 61 62 63 64

3 Find the area between y 3x and y x from x 0 to x 3 3 The area is

Identify the exponential function whose graph is shown below 11 10 8 Im Anoia D

point If f g x 2 x 8 and g x x 8 find f x x 1 help formulas

Find the area under the curve y 4 8 over the interval 1 5

Find the area between the curve of y x 2 and the x axis on the interval 1 2 O O O 2 23 4 21 4 15 4 2 24 2

Let I rx f x dx where f is the function whose graph is shown below y 6 4 2 1 a Use the graph to find L R and M 2 4 R M OL is an overestimate O R is an underestimate OR is an overestimate OM is an underestimate OM is an overestimate 2 b Are these underestimates or overestimates of I OL is an underestimate c Use the graph to find T T How does it compare with I T I 3

Set a 0 and b 2 Find the area under f x by selecting the between f x and g x by selecting the f x g x dx option Use the interactive figure to find your answer f x dx 0 Type an integer or decimal rounded to three decimal places as needed 2 g x dx 0 Type an integer or decimal rounded to three decimal places as needed 2 f x g x dx f x dx f x dx option Find the area under g x by selecting the 0 unded to three decimal places as needed fg x dx c g x dx option Find the area LHE Ave offee ofpom 170 up Car Click here to launch the interactiv figure

Find the exact length of the curve y In sec x 0 x 6

Evaluate the definite integral 5 10x 2 dx 3 X

Next we substitute s x S f 1 F t 2 de dy dx 1 3 1 9 3x 2 into the equation for the arc length function and simplify 1 9t dt 1 2 1 27 3 1 2 9 Step 3 We have found the integral that defines the arc length function To finish we evaluate the definite integral s x t dt dt

Solve the following initial value problem d s 2 dt S 9cos 3t s 0 600 s 0 0

Consider the following curve y 2x 2 0 x 4 3 Set up an integral that can be used to find the exact length of the curve L Find the exact length of the curve dx