1. Determine centroid of the shaded area shown in figure with respect to given reference axes. All units are in mm and R represents radius. AY 8. 14 R 8

Q: Evaluate ||E y²z² dV, where E lies above the cone o and below the sphere p = 1. = T/3

A: Given, ∫∫∫Ey2z2dV   ....(i)Let x=ρsinϕcosθ y=ρsinϕsinθ and z=ρcosϕAnd we know…

Q: Determine the y.coordinate of ater a diation with a scale tactor of 1.5 centered at the origin O00

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Q: Determine the distance y measured from the x axis to the centroid of the area of the triangle shown…

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Q: Find the centroid of the given region. y = 2-x? y = x A (3/22,45/61) В) (5,15) c) (5/14,38/35)…

A: Centroid is x¯,y¯ , where, x¯ = 1A∫ab xfx-gx dx  and y¯ = 1A∫ab 12 fx2-gx2 dxand A = ∫ab f(x)-g(x)…

Q: Ex.7 The area of the surface ∑ = {(x,y,z) €R® : (x² + ²) < 1, z = 9 + xy} is: (A) - 5 (B) 4개 2 (C)…

A: Using polar coordinates.

Q: Determine the y-coordinate of the centroid of the shaded area. Set a = 11 in., b = 28 in. Answer: y…

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Q: If b = 6 cm, c = 3 cm, h = 6 cm, and a = 2 cm, the centroid of the area about x axis is equal to:…

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Q: y=x², y = 1, y =9 y=9, first quadrant

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Q: Consider the parallelepiped with adjacent edges u = 3i + 4j + k v = i+j+ 4k w = i+ 9j + 6k Find the…

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Q: Consider the parallelepiped with adjacent edges u = 3i + 3j + k v = i+j+ 8k w = i+ 8j + 4k Find the…

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Q: In AKLM, R is the centroid. If RQ=10 find LQ. L N P R K M

A: In ∆KLM The centroid of a triangle is "R". The length of the RQ=10. The objective is to find the…

Q: Let C be the curve y = 6x for 0 S< 1.2. %3D 10 8. 7- 6- 5- 4. 2- 0.5 1 Find the surface area of…

A: Given: Let C be the curve y=6x3 for 0≤x≤1.2

Q: Consider the parallelepiped with adjacent edges u = 2i + 4j + k v = i + j+ 6k w = i + 2j + 5k Find…

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Q: 4. Locate the centroid (x, y)of the following composite areas: -3 ft 3 ft I 1.5 ft 1 ft

A: Consider the given composite area and divide then into four areas as shown in diagram below,

Q: Find the surface area of a sphere with radius, r=8 in. S.A. = in (Type an exact answer in terms of…

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Q: Consider the parallelepiped with adjacent edges u = 5i + 8j + k v = i+j+ 9k w = i+ 3j + 5k Find the…

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Q: 17. Evaluate the integral: S, 9x? – 6dx. 18.Determine the center of the circle x² +y² = 25. %3D

A: (17)  Given that I=∫-159x2-6 dx It is known that ∫xndx=xn+1n+1 So, I=9x33-6x-15 =3x3-6x-15 Now we…

Q: 24. Round to the nearest tenth if necessary. 1 13. Point P is the centroid for AMNO. Calculate RP if…

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Q: Consider the parallelepiped with adjacent edges u = 6i + 5j + k v = i+j+ 6k w =i+ 3j+ 6k Find the…

A: Given the adjacent edges of the parallelopiped are u = 6i + 5j + k v = i + j + 6k w = i + 3j + 6k

Q: Question L. Evaluate the following Sffela?+y+2y"dv, where Q is bounded by the hemisphere z = 4-x2-…

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Q: Consider the parallelepiped with adjacent edges u = 4i + 5j + k v = i+j+9k w = i+ 2j + 6k Find the…

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Q: Consider the parallelepiped with adjacent edges u = 8i + 2j + k v = i+j+ 2k w = i+ 3j + 4k Find the…

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Q: IR, US, and OB are all medians of A BRS, and T is the centroid. IR = 2.4 in., BT = 1 in., UT = 0.7…

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Q: 18. Use spherical coordinatcs Evaluate dV,where E is the portion of the unit ball X ty +z²<! that…

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Q: items 12-15, consider the region bounded by the curve y = cosx and the x-axis, where 0 ≤ x ≤ What is…

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Q: 10/ Compute S Fids, uhere S is the surface oriented up ward and E

A: According to divergence theorem ∫∫SF.ds=∬∫Vdiv F dxdydz Given F=y, -x,…

Q: Consider the parallelepiped with adjacent edges u = 9i + 5j + k v = i+j+ 3k w =i+ 6j + 4k Find the…

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Q: Consider the parallelepiped with adjacent edges u = 4i + 6j + k v = i +j+ 9k w = i + 7j + 6k Find…

A: Given; u=4i+6j+k , v=i+j+9k , w=i+7j+6k Volume (V)=?

Q: y 2 2 2 4

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Q: : The area of the surface obtained by rotating the curve y = 2 In a, for 1 < a <e, about the a-axis,…

A: We have to find the integral that given the surface area.

Q: Locate the centroid of the plane area shown. 20 in. Ans: 9.00 20 in- Fig. P5.12

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Q: Consider the parallelepiped with adjacent edges u = 4i + 3j + k v = i+j+ 4k w = i+ 9j + 2k Find the…

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Q: Y 60 cm 40 cm 15 cm +45 cm 15 cm Figure(1)

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Q: 16. Ris the circle sector bounded by x + y = 25 in the first quadrant; 6(x, y) = (vx² + y² + 1)kg/m²…

A: Introduction: The formula of the mass of the lamina with density function δ(x,y) on the region R is…

Q: Consider the parallelepiped with adjacent edges u = 3i + 8j + k v = i+j+ 3k w = i+ 9j + 4k Find the…

A: u=3i+8j+kv=i+j+3kw=i+9j+4kVolume = (u×v).wu×v=ijk381113u×v= i(8(3)-1)-j(3(3)-1)+k(3-8)u×v=23i-8j-5k

Q: Evaluate I= e 2x/y dA, where D is the rectangular region described by 0<x<4 In (10), and 4sy<8.…

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Q: ...) The figure below shows contours of g(x, y) on the region R, with 9 < x < 15 and 2 < y < 8. 7 6…

A: Consider the provided question, R=x,y| 9⩽x⩽15 and 2⩽y⩽8Using ∆x=∆y=2, break the region R into 9…

Q: 3 m 7) Determine the location of centroid (x, ỹ ) of the cross-sectional area shown (P9-69) 1.2 m 3…

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Q: (b) Find the volume and the surface area of the truncated cone made by the rotation of the solid…

A: Formula of cone: Volume of the cone, V=πr3h3 Surface area, A=πrs+πr2 where, r= radius of the base h=…

Q: Locate the centroid of the plane area shown if a 25 in. 20 in. r = 15 in. 30 in.

A: Area of Rectangle equals to : Arectangle=20+3025                =1250in2 Area of semi circle equals…

Q: Let be s a surface compaund by the union of s, qnd Sz An expression to get the Area of the Surface…

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Q: 3: Find the surface area of the part of the paraboloid z = and above the first quadrant. 196 - x -y…

A: The surface area of paraboloid is defined as the sum of the area of all the faces of paraboloid. it…

Q: 3. If D is the centroid of APQR, PA = 17, BD = 9. and DQ = 14, find each missing measure. %3D P a)…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: In ADEF, C is the centroid. E If DM = 48, find DC and CM L M %3D units

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Q: (b) Find the volume and the surface area of the truncated cone made by the rotation of the solid…

A: From the given graph: Now, from the graph given that:    r = 2 unit, R = 4 unit, and h = 3 unit.…

Q: Prove that the surface area of a sphere with radius r is given by S = 4r? using the methods of this…

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Q: Consider a Möbius strip cut along the dashed blue lines as shown below. What is the resulting…

A: When a mobius strip cut along these lines we get ,

Q: 1) A) Find the area of the portion of the sphere x? + y + =9 lying inside the cylinder x2 + y? = 3y.

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Q: 17. Let AXYZ be circumscribed on O R, and let XS = 5, SY = 32, and YZ = 64.* *Image not set to…

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Q: Let A be the area of a circle with radius r. If dA when r = 3. dt dr 3, find dt

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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.