Let S be the solid obtained by rotating the region shown in the figure about the y-axis. (Assume a = 12 and b = 4.) y = a sin(bx)

Q: 3 Through polar coordinates, evaluate S S cos(x² + y²)ā dA, R where R is the region enclosed by…

A: The given question is related with evaluation of a double integral. We have to evaluate ∬R cosx2 +…

Q: /| 2rdV, where E is the region under the plane 2r + 3y + z = 6 that lies in the first octant as…

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Q: SSL. xy dV, where T is the first-octant solid bounded by the coordinate planes and the upper half of…

A: Here we have to find  ∫ ∫ ∫T xy dV , where T is the first-octant solid bounded by the coordinate…

Q: When transforming 9-x-y I LLE (1² + y² + z°}dzdydx ¸ the triple integral 2n n 3 !!! to spherical…

A: Let's find.

Q: Draw the polar curves rị =1– sin 0 and r2 = 1. Set up an integral %3D to find the area of the region…

A: Given: r1=1-sinθr2=1 To Find: i. Draw the polar curves.ii. Set up the integral to find the area of…

Q: Find the area of the region that lies inside both polar curves r = 2 and r = 4 cos 0.

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Q: # 3. Sketch the region R inside the curve r = 1+ cos 0 and outside the circle r = 1 and use polar…

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Q: Find the v o l um e generated by rotating the region bounded by the curves y = ex, x = 0, y = pi abo…

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Q: x(t) cos(t), y(t) = 1+ sin(t), with 0 < t < 2n

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Q: Let S be the solid obtained by rotating the region shown in the figure below about the y-axis. y 12…

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Q: tIND THE AZEA OF THE RECION BOUNDED BY ムラ日ラ0 AND THE THE POLAR X-AXIS AY

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Q: Find the surface area of the solid generated by revolving the cycloid x= a(0- sin g). y =a(1-cos ).…

A: We have to find the surface area of the solid generated by revolving the following cycloid about the…

Q: II| (x + y + z) dv, where E is the solid in the first octant that lies under the paraboloid z = 16 -…

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Q: 6. Determine the volume V of the 3 dimensional lampshade formed (see diagram below) by rotating the…

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Q: 3. Evaluate // (22 + y) dV where G is the solid in the first octant bounded by r = y? and r+z = 1.

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Q: Compute the volume of the solid below 2 – a? – y? and above the region D in the xy-plane. The…

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Q: (a) Given a surface function, z(x, y) == + xy with x cos0 and y= sin 0, 2 %3D az and əz (i) find ax…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Compute the volume of the solid below 2 – a? – y² and above the region D in the xy-plane. The…

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Q: Consider the polar curves C₂: r = 4 cos 0 and C3 : r = 4 sin 0. (a) Determine the slope of the…

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Q: 3. Use polar coordinates to evaluate the double integral. (a) ff sin(x² + y²) d4, where R is the…

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Q: 1. Find the volume of the solid bounded by the paraboloids z = 6- x² - y and z = 2x? + 2y2.Use polar…

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Q: 11. Find the circulation of F(x,y) =(x-y)i+xj around the unit circle centred on the origin. y

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Q: Which of the following is the double polar integral that gives the volume of the solid surrounded by…

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Q: (b) Let E = {(x, y, z) : –1< # < 0, 0 < y < VI– a², 0< z< 2}. Write the integral II| cos (2² + y°)dV…

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Q: Use cylindrical coordinates. Evaluate |/| 6(x3 + xy²) dv, where E is the solid in the first octant…

A: We have to evaluate the integral ∬∫E6x3+xy2dV, where E:z=4-x2-y2 in the first octant.

Q: 10. Let D be the solid inside the cylinder r= cos , above the xy-plane and under the paraboloid…

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Q: 2. Use polar coordinates to find the volume of the solid under the paraboloid z = x² + y² and above…

A: Let's find volume of solid by using polar coordinates.

Q: Let R be the region enclosed by the polar curve r (0) = 2 - 2 sin(0), as shown in the graph. y 1 1 R…

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Q: Consider the polar curve C1 given by r = sin(0), for 0 e [0, 27T), Find the area of the region under…

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Q: Given !! y?Vx² + y² dy dx -1 -V1-x2 a) Sketch the region and rewrite the integral in polar…

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Q: Let x = u cos³ (v), y = u sin³ (v) (a) Sketch the region D in the xy-plane corresponding to 1 < u <…

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Q: Use polar coordinates to evaluate the double integral 2 1 – a²- dy dx. (1+æ? + y²)²

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Q: When transforming the triple integral V9-x2 (x2+y2 +z?) dzdvdx 2n n 3 to spherical coordinates…

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Q: Let S be the solid obtained by rotating the region shown in the figure about the y-axis. (Assume a =…

A: Given data: *The graph is f(x)=y=asin(bx2). *The value of the constant a and b is a=8 and b=4. *The…

Q: Sketch the region and find its area. - The region between y = sin z and y = sin 2r, for 0< I<T.

A: Given: The regions y=sin x and y=sin 2x Consider the two curves are equal, sin x =sin 2x  sin 2x -…

Q: Consider the solid D that lies inside the cone z = and bounded by the spheres 3 y رد جر+ e][] p = 2…

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Q: The volume of the solid generated by the curve y = e 2, between z = 2 and z = 6 about the a - aris…

A: Solution-

Q: Evaluate II| 6x(x2 + y2) dv, where E is the solid in the first octant that lies beneath the…

A: Since the integral lies in first octant so 0≤θ≤π/2 and z≥0.  Converting into cylindrical coordinates…

Q: Evaluate the double integral cos V x2 + y?dA, where D is the disc with center the origin and radius…

A: See the details solution in below

Q: Consider the polar curves C₂: r = 4 cos 0 and C3 : r = 4 sin 0. (a) Set up the integral that gives…

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Q: Use cylindrical coordinates. Evaluate 3(x3 + xy2) dV, where E is the solid in the first octant that…

A: Given: The solid E is in first octant that lies beneath the paraboloid z=1-x2-y2.

Q: The volume of the solid generated by the curve y = e", between a = 2 and a = 6 about the a – aæis is…

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Q: Q1/ Find the volume of the solid generated by revolving the shaded region about the x- axis for the…

A: To find:- Volume of solid  generated by revolving the shaded region about x axis by disc method…

Q: Let R be the region inside the ellipse x = 4 cos 0, y = 3 sin and outside the circle x = 2 cos 0, y…

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Q: Find the area of the surface formed by revolving about the polar axis the following curve over the…

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Q: Consider the polar curves C2 : r = 4 cos 0 and C3 : r = 4 sin 0. (a) Determine the slope of the…

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Q: 7. Use spherical coordinates to find ||| z dx dy dz where E is the solid defined by E Os Vr* +y³ <z…

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Q: Consider the polar curves C2 :r = 4 cos 0 and C3 : r = 4 sin 0. (b) Set up the integral that gives…

A: The given curves are C2: r=4cosθ and C3: r=4sinθ .........(1) To find: The area under the given…

Q: Let S be the solid obtained by rotating the region shown in the figure below about the y-axis. y…

A: The function of the curve is y = 6x(x-1)2.

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.