Definite Integrals Questions and Answers

4 a If x y 25 find by implicit differentiation dy dx b Find an equation of the tangent to the circle x y 25 at the point 3 4 3 3

Cylindrical z r cos 0 y rsin 0 2 2 dV r dr d0 r 2 y Spherical z pcos sind y psin sind z pcos dV p sine dp do do p 1 y 1 10 points Find the volume bounded by the cylinder 2 y 4 and the planes z 0 and y z 3

5 Find Love 1 x dx a interpret it geometnically

4 Find the arc length of the curve 3 4 for Y y 1 x 8 8 pts 5 Find the area of the surfaces generated by revolving the curve x4 3 about the Y axis for 3 2 3 8 x 3 5 7 17 2 2 2 3

y rsin z z dV r dr d0 r y Spherical z pcos sino y psin sind z pcoso dVp sine dp do do p 1 y 2 2 10 points Evaluate y 2 dV where E is the solid hemisphere that lies above the zy plane centered at the origin with radius 1

7 6 1 2 0 8 0 47 H ree o 1 201 Homework 1 2 1 Sketch the region enclosed by the curves and find its area X 114 a y x v x x 0 4 c 0 8 y x y x 2 x 1 x 1 b y x 4x y 0 x 0 and x 2 d y x y x 2 g x y 1A 2 1 Sy 3 3 1 49 192

Find the volume of the solid that results when the region above the x axis and below the ellipse 3 V a b 1 is revolved about the x axis

Determine the values of x and y such that the points 1 2 3 6 7 1 and x y 2 are collinear lie on a line X and y Type integers or decimals

Find the average value for the function fave over the given interval Then find c a b such that f c fave 3 f x 31 9x 5 4 g x 3 on 0 1 on 1 1

Evaluate f x y z dV for the function f and region W specified 12 x y dV f x y z 12 x y W y z x 0 y x 0 x 1

Integrate f x y z 20xz over the region in the first octant x y z 20 above the parabolic cylinder z y and below the paraboloid z 8 2x y 2

Let h x e 5 Decompose h x into two functions so that h x f g x f x g x Note In order to get credit for this problem all answers must be correct

Find the volume under the paraboloid z x 4y and above the rectangle 0 2 1 4

1 point Consider the solid that lies above the square in the xy plane R 0 2 x 0 2 and below the elliptic paraboloid z 49 x 3y Using iterated integrals compute the exact value of the volume

3 6 points Using polar coordinates find the volume of the solid below the cone z y and above the ring 1 x y 25

Evaluate the iterated integral 6 x 4y dx d

1 point Using geometry calculate the volume of the solid under z 1 x and over the circular disk x y 1

of SL ve the hyperbola xy 1 and the line y x and below the line y 2 7 points Calculate the value of y dA where D is the region in the first quadrant that lies above

F x dx 3 and t If g x dx 3 what is the value of f xx y dA where D is the rectangle 1 x 3 2 y 2

1 point Evaluate the integral by reversing the order of integration With order reversed where a b C Evaluating the integral cos x dx dy So I co cos x dx dy co cos x dy dx and d

1 point Find the mass of the triangular region with vertices 0 0 3 0 and 0 1 with density function p x y x y

sketch the region of integration and evaluate the integral Your sketch should be approximately the same as one of the graphs shown below which is the correct region Graph Then e dy dx Note the value of the integral needs to be correct to two decimal places Graphs exp 9 3 X 3 X To ex dydx J 2 5 ton X X exp 9 co X x

x 1 point Evaluate the double integral I xy dA where D is the triangular region with vertices 0 0 2 0 0 5

N 16 x sketch the region of integration and evaluate the integral Your sketch should be approximately the same as one of the graphs shown below which is the correct region Graph Then 2 16 x 2xy dy dx Graphs 5 2xy dy dx 2 6 2

1 point Suppose R is the shaded region in the figure and f x y is a continuous function on R Find the limits of integration for the following iterated integrals a f x y dA f x y dy dx R A B C D b fff x y dA f x y dx dy E F G B H R

1 point By changing to polar coordinates evaluate the integral where D is the disk x y 16 t tri dody

point Match each double integral in polar with the graph of the region of integration 3x 4 So Lot 10 1 123 0 2 0 6 0 1 3 4 5 2x 1 fr 0 r dr do fr 0 r dr de 14 2x 1 fr 0 r dr do 3x12 fr 0 r dr do 6 7 3x 4 8 3x12 3x14 1 8 1 32 f r 0 r de dr f r 0 r de dr f r 0 r dr de 3 4 f r 0 r de dr G A O D G D B E D H C

1 point Suppose R is the shaded region in the figure As an iterated integral in polar coordinates with limits of integration A B D f x y dA R B C D f r cos 0 r sin 0 r dr de 2 1 Click on graph to enlarge

1 point Find the mass of the rectangular region 0 x 4 0 y 3 with density function p x y 3

Define f x g t dt where the graph of g t is shown here 5 0 5 5 a Find the value of f 2 b Give the lower Riemann sum approximation of f 2 using a partition with At 2 10

Let R be the region in the ry plane bounded by y 2 and the z axis which is shown in the figure below y Q10 1 2 Points O 2 1 0 1 da Which of the following represents the volume of the solid obtained by rotating R around the 2 axis dz 14 dz 2 30

What is the resulting integral if the substitution v is applied to da 2 edu 2fe du 2 d Suntstelle WRANGLg star PERICU

Find the exact value of csc P in simplest form 2 P 3 13 N

If f f x dx 10 and f f x dx 4 find f f x dx 6 O None of the above O 10

2 15 18 Question 6 1 point Listen Express CA TC RC RT TC RC R CA RT as a single vector

Find the area of the region R bounded by the graph of f x 12 x and the x axis on the interval 12 24 Graph and show the region R Choose the correct graph below The region R is shown in yellow A 48 Ay The area of Ris 48 OB 48 48 48 O Q C 48 48 48 48 Q Q S OD 48 48 48 Q S

Find the net area and the area of the region bounded by y 5 cos x and the x axis between x the function and find the region indicated in this question The net area is Simplify your answer K N 2 and x Graph

Estimate the area of the region bounded by the graph of f x x 4 and the x axis on 0 10 in the following ways a Divide 0 10 into n 4 subintervals and approximate the area of the region using a left Riemann sum Illustrate the solution geometrically b Divide 0 10 into n 4 subintervals and approximate the area of the region using a midpoint Riemann sum Illustrate the solution geometrically c Divide 0 10 into n 4 subintervals and approximate the area of the region using a right Riemann sum Illustrate the solution geometrically a The left Riemann sum is Type an integer or decimal rounded to three decimal places as needed

Find the area of the region bounded by the graph of f x x x 2 x 1 and the x axis on the interval 2 The area is Simplify your answer

Find the area of the region bounded by the graph of f x sin x and the x axis on the interval x 3 3 4 The area is Type an exact answer using radicals as needed www

5 Consider the surface begin array 1 x u v 2 u 3 v y u v 2 u 3 v z u v u v end array where u 0 1 and v 0 2 Calculate the area of this surface

Sketch the region enclosed by x y 30 and x y 0 a Favoring convenience should you integrate with respect to z or y b What are the limits of integration lower limit and upper limit c Find the area of the region by integrating

Given Rn 2 2 1 2 cos 72 a that is provide a b and f z in the expression b f f express the limit as n 0 as a definite integral f x f x dr

A faucet leaks 6 oz of water per minute a How many gallons of water are wasted in a year A gallon contains 128 oz b If water costs 9 72 per 1000 gal how much additional money is being spent on the water bi a gallons would be wasted in one year Tyne an integer or decimal rounded to the nearest hundredth as needed

4 Use L H pital s Rule to prove that the function 5 F x Lea ta dt is continuous for all values of x

1 6 points Find the length of the curve r t 2t t t for 0 t 1

valuate the related series of each sequence 34 40 46 52 58 64

Evaluate the function at the specified points f x y y xy5 5 4 3 1 1 4 At 5 4 At 3 1 At 1 4

Let f x Can we use the mean value theorem to say the equation f x 2 x 4 1 x Choose 1 answer A B No since the function is not differentiable on that interval 1 4 has a solution where No since the average rate of change of f over the interval 2 x 4 isn t equal to 1 4 Yes both conditions for using the mean value theorem have been met

What is the value of log279 0 13 1 2 O NO CYCO 2