Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln 3x, y = 2, y = 3, x = 0; about the y-axis Sketch the region.
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. x = 2 7y, x = 0, y = 3; about the y-axis V = Sketch the region. Sketch the solid, and a typical disk or washer.
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 3 9−x2, y = 0, x = 0, x = 2; about the x-axis V = Sketch the region. Sketch the solid, and a typical disk or washer.
Let R be the shaded region in the first quadrant enclosed by the graphs of y = 1 + sinx y = ex2 − 1 , and the y-axis, as shown in the figure below. *Hint: the functions do NOT intersect where x = 1. Find the point of intersection. A) Find the area of the region R. B) Find the volume of the solid generated when the region R is revolved about the x-axis. C) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid.
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 3 sin(x), y = 3 cos(x), 0 ≤ x ≤ π/4; about y = −1 V = Sketch the region. Sketch the solid, and a typical disk or washer.
Find the centroid (center of mass) of the following thin plate, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify the work. The region bounded by y = lnx, the x -axis, and x = e10 The centroid is located at. (Type an ordered pair. Type an exact answer.)
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 27x3, y = 0, x = 1; about x = 2 Sketch the region. Sketch the solid, and a typical disk or washer.
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. V = y2 = 2x, x = 2y; about the y-axis Sketch the region. Sketch the solid, and a typical disk or washer.
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. y = 5e1−x2, y = 5x4 a) 3.66016 b) 91.504 c) 18.3008 d) 3.141257 e) 12.08127
The graph above shows the base of an object. Compute the exact value of the volume of the object, given that cross sections (perpendicular to the base) are equiangular triangles. V =
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 7x2, y = 7x, x ≥ 0 V =
Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 4 − 4x2, y = 0 V = Sketch the region.
Find the area of the region bounded by the equations by integrating (i) with respect to x and (ii) with respect to y. x = 16−y2 x = y−4 A = 727 12 A = 365 12 A = 243 4 A = 2432 A = 365 6
A rectangle is to be placed in the first quadrant, with one side on the x-axis and one side on the y-axis, so the rectangle lies below the parabola y+3x = 4. What is the maximum area of this rectangle? a) 43 b) 2 c) 23 d) 3
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 8. y = sinx, y = 0, 0 ≤ x ≤ π2 π(16−π4) π(32−π4) π(16−π2) π(8−π2) π(8−π4)
The region R in the first quadrant of the xy-plane is bounded by the curves y = −3x2 + 15x + 18, x = 0 and y = 0. A solid S is formed by rotating R about the y-axis: The (exact) volume of S is: Preview The region R in the first quadrant of the xy-plane is bounded by the curves y = −6sin(x), x = π, x = 2π and y = 0. A solid S is formed by rotating R about the y-axis: The volume of S is: Preview
Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by y = x9 and y = 512 in the first quadrant about the y-axis. V = π∫09 y29 dy = 18, 43211π V = π∫0512 y19 dy = 9, 21611π V = π∫09 y29 dy = 9, 21611π V = π∫0512 y29 dy = 1843211π V = π∫0512 y29 dy = 9, 21611π
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 6. y = x, y = 5, x = 0 1003π π 2753π 2756π 2003π
Find the area of the region bounded by the given curves. y = sin(πx 10), y = x2 − 10x a) 16+20π b) 5003−20π c) 5003+20π d) −850+π20 e) None of these
For the region in the first quadrant bounded by y = 1 − x2, the x-axis, and y-axis, determine which of the following is greater: the volume of the solid generated when the region is revolved about the x-axis or about the y-axis. When the region is revolved about the x-axis, the volume is . (Type an exact answer, using π as needed.) When the region is revolved about the y-axis, the volume is. (Type an exact answer, using π as needed.) Which volume is greater? A. The volume about the x-axis is greater. B. The volumes are equal. C. The volume about the y-axis is greater.
Find the mass and centroid (center of mass) of the following thin plate, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. The region bounded by y = lnx, the x-axis, and x = e Identify the centroid (center of mass). (Type an ordered pair. Type an exact answer.) Sketch the region, identifying the centroid with a red dot. Choose the correct graph below. A. B. C. D.
Find the area of the region in the first quadrant bounded by the line y = 5 x, the line x = 5, the curve y = 5/x, and the x-axis. First sketch a graph of the region. Choose the correct graph below. A. B. C. D. Set up the integral(s) that will give the area of the region. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. ∫0()dy+∫5()dy C. ∫()dy B. ∫0()dx+∫5()dx D. ∫()dx
Let g(x) = 10x + sin10 x for 0 ≤ x < 2π. The function g will have critical points in the interval [0, 2π). List in increasing order the first two critical points in [0, 2π). Enter DNE for any empty answer blank. Type pi or use the calcPad symbol for π if needed. x = x = symbolic formatting help
Suppose that R1 = 7.41 + 0.6t and R2 = 7.41 + 0.38t model the revenue (in billions of dollars) for a large corporation. The model R1 gives projected annual revenues from 2008 through 2015, with t = 8 corresponding to 2008, and R2 gives projected revenues if there is a decrease in the rate of growth of corporate sales over the period. Approximate the total reduction in revenue if corporate sales are actually closer to the model R2. Round your answer to three decimal places. $10.780 billion $157.780 billion $1.540 billion $24.010 billion $17.710 billion
Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The coordinate axes and the line 2x + 2y = 2. Choose the correct graph of the region. A. B. C. D. Express the shaded area as an iterated integral. ∫∫dydx Evaluate the integral from the previous step. The value of the integral is.
A solid right (noncircular) cylinder has its base R in the xy-plane and is bounded above by the paraboloid z = x2 + y2. The cylinder's volume is V = ∫01∫0 y(x2 + y2)dxdy + ∫12∫02−y(x2 + y2)dxdy Sketch the base of the region R and express the cylinder's volume as a single iterated integral with the order of integration reversed. Then evaluate the integral to find the volume. Choose the correct sketch of the region R in the xy-plane. A. B. C. D. Express the cylinder's volume as a single integrated integral. ∬(x2+y2)dydx Enter your answer in the edit fields and then click Check Answer. ?
Find the centroid (center of mass) of the following thin plate, assuming constant density. Use symmetry when possible to simplify the calculations. The region in the first quadrant bounded by x2 + y2 = 25. The coordinates of the centroid are (x¯, y¯) = . Type an ordered pair. Type an exact answer, using π as needed.)
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations y = 4x2, y = 0 and x = 2 about the line x = 2. 16 3π 32 3π 128 3 128 3π 32 3
Sketch the region bounded by the curves y = 7x3, y = 7 and x = 0 then find the volume of the solid generated by revolving this region about the x-axis. a) 41π b) 46π c) 42π d) 44π e) 45π
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. x = 7 − 7y2, x = 7y2 − 7 Find the area of the region.
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 7x5, y = 7x, x ≥ 0; about the x-axis V = Sketch the reaion. Sketch the solid. and a tvoical disk or washer.
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. 6x + y2 = 16, x = y Find the area of the region.
For the given double integral, sketch the region of integration and write an equivalent double integral with the order of integration reversed. ∫01∫−1−y2 1−y2 5y dxdy Sketch the region of integration. Choose the correct graph below. A. B. C. D. What is an equivalent double integral with the order of integration reversed? ∬5 dydx
Sketch the region enclosed by the given curves. y = 7 cos(2x), y = 7 − 7 cos(2x), 0 ≤ x ≤ π/2 Find its area.
Sketch the region bounded by the curves x = y2 and 2x = y+1 then use the shell method to find the volume of the solid generated by revolving this region about the y-axis. а) 49π 20 b) 9π 20 c) 29π 20 d) 89π 20 е) 69π 20
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 8x3, y = 0, x = 1; about x = 2 Sketch the region.
The region in the first quadrant bounded by x2 + 4y2 = 64, the x-axis, and the y-axis is revolved around the y-axis. The volume of the resulting solid is 176π 3. 256π 3. 512π 3. 1,024π 3
The integral gives the area of the region in the xy-plane. Sketch the region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. ∫0 12 ∫ y2/6 2y dxdy Choose the correct answer below. A. B. C. The area of the region is. (Simplify your answer.)
Use a computer algebra system to find the exact volume of the solid. between the paraboloids z = 2x2 + y2 and z = 9 − x2 − 2y2 and inside the cylinder x2 + y2 = 1
Consider the following functions. f(x) = 2 sinx g(x) = 2 cos2x −π2 ≤ x ≤ π6 Sketch the region bounded by the graphs of the functions. Find the area of the region. (Round your answer to three decimal places.)
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y2 = 2x, x = 2y; about the y-axis V = Sketch the region. Sketch the solid, and a typical disk or washer.
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = 1 7x+9 y = 0 x = 0 x = 6
The graph of f(x) = 6e−7x, the x-axis, the y-axis, and the line x = 3 form a bounded region. Find the volume of the solid obtained by revolving the region about the y-axis. Select the correct answer below: 127π(1−4 e−21) units 3 127π(1+4 e−21) units 3 1249π(1−22 e−21) units 3 1249π(1+22 e−21) units 3
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 8. y = x, y = 0, y = 7, x = 8
Find the area of the region bounded by the graphs of the equations. f(x) = sin(2x), g(x) = cos(x), −π2 ≤ x ≤ π6 A = 94 A = 9 3/22 A = 3 3/22 A = 3 3/28 A = 9 3/28
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the y-axis. y = x, y = 7x, y = 21 The volume of the solid is cubic units. (Type an exact answer.)
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. (Do this on paper. Your instructor may ask you to turn in this graph.) y = 2x2, y = 6x − x2 Then find the area S of the region. S =
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. (Do this on paper. Your instructor may ask you to turn in this graph.) 2x + y2 = 15, x = y Then find the area S of the region. S =
Use the method of cylindrical shells to set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. a) x = 9, x = y + 14/y; about the the line y = −2.
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = 5/x y = 0 x = 1 x = 5
Find the centroid (center of mass) of the following thin plate, assuming constant density. Use symmetry when possible to simplify the calculations. The region in the first quadrant bounded by x2 + y2 = 289 Make a sketch of the region. Choose the correct answer below. A. B. C. D. Set up the double integral that gives My, the plate's first moment about the y -axis. Use increasing limits of integration. Assume a density of 1. My = ∫()dydx (Type exact answers.) A. This integral is not necessary because the given function is symmetric about the y-axis. B. This integral is not necessary because the given function is symmetric about the line y = x. C. This integral is not necessary because the given function is symmetric about the x-axis. Dhis integral is necessary; Mx = ∬(dydx (Type exact answers.) The coordinates of the centroid are (x¯, y¯) = (Type an ordered pair. Type an exact answer, using π as needed.) Plot the centroid on the sketch of the region. Choose the correct answer below. A. B. C. D.