Integratef (x, y) = xy over the region R, which is a semi-circle. 4 R 8 Enter the exact answer. f (x, y) dA =

Q: b) 1: Find the volume of the solid obtained by rotating the region bounded by the curve about the…

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Q: evaluate triple integral 2z dv, where E is the solid hemisphere x^2+y^2+z^20

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Q: 6 x y . Evaluate the double integral /JR Va? + y² + 1 dA, over the rectangular region R = {(x, y) :…

A: Given : Double integral ∫∫R 6xyx2 + y2 + 1dA Rectangular region R = x,y : 0≤x≤1, 0≤y≤1   To find :…

Q: Q.4. Find the volume of the solid that results when the region enclosed by the curve y3sinx, y = x…

A: Volume problem....

Q: ind the volume of the resulting solid if the region under the curve  y = 8/(x2 + 3x + 2)  from x =…

A: Consider the curves y = 8x2+3x+2, x=0, x=1 about x-axis For this region of the disk is y = 8x2+3x+2…

Q: t) The volume of the solid obtained by rotating the region enclosed by y = x', y = 25x, x > 0 about…

A: Given

Q: y=x2-2x+1 and y= -x2+2x+7   Use washer method to find the volume of the solid whih is obtained by…

A:

Q: Q3. Find the volume of the solid obtained by rotating about the y-axis the region between y = 0 and…

A:

Q: 9. Find the area of the region enclosed by the curve x = y? – 4y and x = 2y – y². (Hint: integrate…

A: Area between  two given curves

Q: Calculate the line integral x y dx + x² y³ dy where C is the triangle with vertices (0,0), (1,0),…

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Q: 11 Consider the integral f(x, y)dyda. Sketch the region of integration and change the order of…

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Q: Ql: Find the volume of the solid generated by the region bounded by y= Vx and the lines y- 1, x 6 by…

A: We will find the volume as following.

Q: Find volumc of the solid gcncrated the by revolving about the y-axis the region in 4y-y and 4-y x =

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Q: Q. N. 3 Find the volume of the solid generated from revovling the area of the region bounded by the…

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Q: PTODIEI4 TIII 1. dA y3 +1 where D is the region bounded by y = 2, r = 0, y = Vx.

A: Consider the provided integral, The region D bounded by the given equation is draw as,

Q: 64z dV where V is the region below x² + y² + z² = 1 and inside Evaluate Væ2 + y². Answer:

A:

Q: -1-y 2. The triple integral S, S2 S, dz dy dx gives the volume of the region shown below.

A: To determine the value of the triple integral.

Q: 2 S[(y-ýj-y] dy D Draw the and calculate integral region of integral your value

A:

Q: 1. Find the area of the region bounded by the parabola x = 3+ 2y - y? and the line 2x + 3y = 6. Find…

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Q: Evaluate the double integral I3 = SS (22-2 3) dxdy where S is the region in the 4th quadrant (y < 0,…

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Q: 5. V is a volume of the solid obtained by rotating about line x = 1 the planar region bounded by…

A:

Q: f(x) = x³, y = 1, x = 1 Find the volume of the solid obtained by rotating the region A about the…

A: Solution :

Q: 5x dV, where E =    (x, y, z) | 0 ≤ y ≤ 2, 0 ≤ x ≤  sqrt4-y^2, 0 ≤ z ≤ 6y   evaluate triple…

A: Answer is 60

Q: f(x) = -x2 + 4. Calculate the volume of the solid generated by rotating the region between f(x), y =…

A: We graph the equations using a graphing calculator.  The shaded region is rotating about the y=5…

Q: 4. Consider the region in the first quadrant bounded between the x-axis, the y-axis, and the curve…

A: Consider the provided question,  Consider the region in the find quadrant bounded between the x-axis…

Q: JJXVX+5y dA, where R = {(x,y): 0sx55, - sys4 – x>; use x= 5u, y = 4v – u. R a. Sketch the original…

A:

Q: 8. (a) Find the area enclosed by the curves y = x2 – 4 and y (b) Find the volume of the solid that…

A: (a).  The area between the given curves is: Area=∫-172 5x2-12-x2+4 dx

Q: P3: Calculate the volume of solid of revolution when the region bounded by y = cos x and x² - y - 2…

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Q: b) [ bounded 1: Find the volume of the solid obtained by rotating the region by the curve about the…

A: Graph:

Q: (b) Find the volume of the resulting solid if the region under the curve 1 from x = 0 to x = 1 is…

A:

Q: x +y over the solid region bounded by z = 0 and z = 25 - Intergrate Give your answer as a reduced…

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Q: Evaluate the triple integral |// - + 47x²yz²)dV, where V is the cylindirical region x² + y? < 4,0 <…

A:

Q: I V9+y? dy dæ by first interchanging the order of integration. Sketch the region of integration.…

A: Order of change of integrals

Q: Use Green's Theorem to evaluate the line integral. 2xy dx + (x + y) dy C: boundary of the region…

A: We need to evaluate integral using Green's theorem.

Q: Q 1/ Find the volume of the solid that is bounded above by the cylinder f(x,y) = x² and below by the…

A: The given curves are f(x,y)=x2, parabola y=2-x2 and the line y=x To evaluate the volume we will use…

Q: me volume ốf the solid obtained by rotating the region enclosed by y = x², y = 4x, about the line y…

A: Find the volume of solid

Q: 1r + 3x + 18, The region R in the first quadrant of the xy-plane is bounded by the curves y = x = 0…

A: Given: The given curve is y=-x2+3x+18,x=0 and y=0.A solid S is formed by rotating R about the…

Q: 07) Find the volume of the solid bounded above by the plane z=4-x-y and below by the rectangle R-…

A:

Q: The surface area of the surface generated by revolving y=x³ , 0<x<1 about the x- axis is S×3.56312…

A: Option 2 none

Q: y=x2-2x+1 and y= -x2+2x+7  Use shell method to find the volume of the solid which is obtained by…

A:

Q: Q2 Evaluate er²+y? dydx R where R is the semicircular region bounded by the x – axis and the curve y…

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Q: P3: Calculate the volume of solid of revolution when the region bounded by y = sin 2x and x²-y-2 = 0…

A:

Q: 3. Find the volume of the solid under the plane 3x+ 2y – z = 0 over the shaded region between the…

A:

Q: f(x) = x³, Y Find the volume of the solid obtained by rotating the region A about the x-axis. A B

A: The region A is bounded by y=1 and y=x3 from x=0 to x=1

Q: Rotate the region bounded by y = 3x*,y=D0, and x=3 aberut 4= 3x2 the granis: Find the resutine…

A: Let the R be the region bounded by y=3x2 and x-axis between x=0 and x=3 .…

Q: 2. Find the volume of the solid obtained whe the region bounded by the curve y = sin(x), 0 < x < ™,…

A: For the given information

Q: raluate the surface integral. xz ds S is the boundary of the region enclosed by the cylinder y2 + z2…

A: S is the boundary of the region enclosed by the cylinder y^2+z^2=9 and the planes x=0 and x+y=8 S1:…

Q: Q3) Find the volume of the solid generated by revolving region bounded by the curve (y=e* ) & the…

A:

Q: ー元 Q.3 Sketch the region enclosed by the curves y = sec' x, y=x² +1, xr=- 4 and find 4. its area.

A: Sketch the region and integrate the function with x from -π/4 to π/4.

Q: Intergrate z+y over the solid region bounded by z 0 and z = 4-a-y Give your answer as a reduced…

A: Given the solid region is bounded by z=0,z=4-x2-y2. Let the solid be E. Find ∭Ex2+y2dV.

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.