The region R in the first quadrant is bounded by y = 10 sin y = 10(x – 2)² and y = 8x + 2, and contains the point (2, 10). The area of R is: area The volume of the solid generated by rotating R about the r-axis is: volume = The volume of the solid generated by rotating R about the y-axis is: volume The volume of the solid with R as its base and square cross-sections perpendicular to the r-axis (and to the ry-plane) is: volume =

Q: find the centroid of the thin plate bounded by the graphs of the given functions. Use d = 1 and M =…

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Q: Rectangular coordinates for the integral rV4-r2 z?p3sin® cos0 dz dr de is V3-x2 4-x²-y² 1- xyz? dz…

A: Consider the integral ∫0π2∫13∫14-r2z2r3sinθcosθdzdrdθ. Find the rectangular coordinates for the…

Q: (4) Find the surface area of the cone S a) b) c) d) 7-√√2 5-√3 p(r, 0)=(r cos 0,r sin 0,r) for 0≤0…

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Q: 7 Find the lateral (side) surface area of the cone generated by revolving the line segment y = 3x,…

A: Given: Cone generated by revolving the line segment y=73x, 0≤x≤8, about x-axis. To Find: Lateral…

Q: 1. Let a be a positive number. Find the surface area of the solid obtained by revolving the curve…

A: We use the disk method to find the volume

Q: 5. Let L be the length of the parametric curve x = ½(t – sin(t)) and y = ½(1 – cos(t)) from t= 0 to…

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Q: Three curves have been drawn polar coordinate, circle, (r= 5), circle; (r=3 sin (0), use double…

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Q: Find the area of the region that lies outside the cardioid r = 1 + cos θ and inside of the circle r…

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Q: 1. Shown in the left side figure, is the differential solid angle d2 subtended by the differential…

A: Asper the question we have to find the total solid angle Ω subtended by the circular disc of radius…

Q: There is an ant on a circular heated plate. The plate temperature is given by f(x, y) = y² – x² + 12…

A: Consider the function fx,y=y2-x2+12, with equality constraint x2+y2=121 (since the radius is given…

Q: 2. Let R be the region below enclosed by the curves z = 2 − y², y = − sin (772), and x = -1, as…

A: Both cases required to divide the region into two subregions.

Q: (a) Calculate the centroid (7, j) of the region under the graph y = cos x for 0 < x <T/2. (b) Show…

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Q: Find the surface area of the solid obtained by revolving the curve y = sin3 t, x = cos3t about the…

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Q: rod with density 8(x) = 4 + sin(x) (in mass per unit length) lies on the x-axis between x = 0 and x…

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Q: a. Find the length of one arch of the cycloid x = a(t - sin t), y = a(1 - cos t). b. Find the area…

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Q: .......n....... ..... Sketch the regions enclosed by the given curves. y = 2 cos(4x), y 2 sin(8x), x…

A: Given problem:- Sketch the regions enclosed by the given curves.   y = 2 cos(4x), y = 2 sin(8x),…

Q: a The integral that represents the length of the curve 7(t) = (2/t, cos (2t3/2), sin (2t3/2)), where…

A: length of curve

Q: 9. Use polar integration to evaluate || cos(x² + y²) dA where R is the region R that lies above the…

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Q: (a) Find the minimum and maximum xcoordinates of points on the cardioid r =1−cosθ.(b) Find the…

A: given,       the cardioid r= 1-cosθ

Q: u(0, t) = u(1, t) = 0 %3D 1 u(x, 0) = - sin(nax) 0 1 >0 0 ) < x < } SI< 1.

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Q: 1. The region inside both the cardioid r =1 – cos 0 and the circle r = 1.

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Q: 2. The arc of the parabola y = x² from (1, 1) to (2, 4) is rotated about the y-axis. Find the area…

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Q: 7. The area of region R bounded by the curve C is given by the line integral P-y dx + x dy 2 By…

A: The area of region R is given by: 12∮C -y dx+x dy

Q: Find the work done by F = 2xy°i + 7x²y²j in moving a particle once counterclockwise around the curve…

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Q: cos 2z dz, C the semicircle |z| = 7, x2 0 from - πί to πί

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Q: Find the area of the surface formed by revolving the circle r = f (0) cos 0 about the line θ = π/2.…

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Q: 29-32 Use double integration to find the area of the plane re- gion enclosed by the given curves.…

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Q: With a curve 56 3 17 x² + y = 6, y = x, y = -2 and x = -2. . Determine its area.

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Q: 5z + sin z TE-31 -dz, C the boundary of the square with vertices+4, ± 4i counterclockwise.…

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Q: Find the centroid of the region bounded by the graphs of the functions 1 and touching the origin. y…

A: Given curve is y=5sinx, y=17x and x=π2. We have to find the centroid x¯,y¯. The formula for the x¯…

Q: 3. Let R be the region below bounded by the parabola y = 4 - 2² and the lines 3x - 2y +3=0 and y =…

A: The region R is bounded by the parabola y=4-x2 and the line 3x-2y+3=0 and y=0.

Q: 2/3 2/3 2/3 2. Find the area of the four-cusped hypocycloid x + y = a 3. Find the area bounded by y…

A: Here we have to find the area enclosed by the given curves. We will use following formulas-…

Q: 1.2. The area of the region bounded by the curves sin r, y = cos r, I= 0 and a equal to A. 1.4142…

A: Given curves are y=sinx, y=cosx ,x=0 and x=π/4

Q: ) feas)= over the region in the 1 quadrant bounded by %3D the hes y=X メニ2. ノY=」

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Q: z = sin(2x) cos(y) 1. Construct a grid for x and y going from (0, 0) to (4, 6). 2. Use this grid to…

A: Using the Geo Gebra software I have plotted all these three problem in 3d graph. 1. The grids are…

Q: The area of the region inside the circle r = 6 cos e and outside the cardioid r = 2(1+ cos 0), a > 0…

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Q: 3. The base of a cone is the unit circle centered at the origin in the r – y plane with vertex P =…

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Q: 7. (Cauchy's integral formula) Compute the integrals (a) [z cos z dz (2² + 1)² 2:2 - 2z +1 (b) where…

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Q: (b) Let R be the region in the positive quadrant bounded by the straight lines 4x, and by the…

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Q: cosh(2z) Integrate counterclockwise around the circle |z| = 8. 25 cosh(2z) dz 25

A: using cauchy residue theorem....

Q: 7 sin(æ), æ = T, I = 27 and The region R in the first quadrant of the xy-plane is bounded by the…

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Q: F(x, y,z) = sin(7x) i+ zy° j+(z² +4x) k and S is the surface of the box with -1<x< 2,0<y<1 and…

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Q: Siupe of the langent iine al any point (X, Y) on the curve 3x2 -2x +7 and the point (1, the curve.…

A: Topic:- application of integration

Q: Approxinale whe enlerral Su Jx cos xdx with n=f use frapezoidal rule G) use sim pron's Rele b)

A: a).   From the given problem:   ∫14xcosxdx ; n=6   As we know that the expression for the…

Q: 3) A curve C is the boundary of a shaded region D which defined by y = 2x² - 4 and y = 2x in…

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Q: 29-32 Use double integration to find the area of the plane re- gion enclosed by the given curves.…

A: Solution is in step 2

Q: 25, (x, Y) 4. A fence is built with posts arranged in the xY-plane along the quarter circle x- +?…

A: Given

Q: 8. Find the area of the region bounded by the parabolas y = 6x – x² and y = x2 - 2x. 9. Find the arc…

A: according to our guidelines we can answer only one question and rest can be reposted.

Q: 2K-tarメ %3D The Extreme of the turcrion tcx = 2x-toanx absolute Extrema [-1っ3,1? 2. on the interva!

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Q: (4– x² – y²) dA Evaluate the double integral JI 2M+3N+2P d) where coefficients M ,N,P are the last…

A: Given   ∬C4-x2-y22M+3N+2P, where M=0,N=2,P=8. Evaluate the integral, here C is a positively oriented…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.