y. (2, 1) 1+ x5 2y x 5 y1 1 1 2 X

Q: SI (1 0x°y - 6)ixdy where S is the region bounded by -2y? and Evaluate X = x = y'.

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Q: Evaluate the integral| c. ff sin x dA; R is the region bounded by the lines y = 2x, y = x and x = .

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Q: 2. Evaluate the following integral where R is the region enclosed by the lines 2x + y = 0, 2x + y =…

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Q: Given the region R on the right, 2 evaluate the iterated double integral y = x .y.x.. dA, where f(x,…

A: Let's find.

Q: Use the substititutions и — у — х, and v 3 y + 2x to evaluate the integral - (2х + у) (х — у) dx dy…

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Q: Find the area of region enclosed by x = t – ť², y = t – t", 0 <t < 1 using Green's Theorem.

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Q: Sketch the region R whose area is given by the iterated integral. *16 – y² dx dy 14 12 10- y 8. 4 2-…

A: Solve by using graph

Q: 10. Find the area bounded by the curve y²-9x and its latus rectum. A. 10.5 C. 11.5 B. 13.5 D. 12.5

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Q: The double integral ff 2xycosydA over the rectangular region R = {(x,y) E R2, 0sxs1 and 0 sysn} is…

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Q: Q:A: Determine the area of bounded by y = e, and y = xv +1 at x-axis. y=zv+1

A: Given areas y=e-x2  &amp;  y=x2 +1xand  xasisso required area =∫-30 e-x2 dx + ∫-30  x2+1x dx…

Q: Let D be the region in R bounded by the curves x = y? and y = x – 2. Which of the following…

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Q: Use an iterated integral to find the area of the region. y = 25 - x2 dy dx = y 25 20 15 10 5 1 1 2

A: The equation of the curve is given by y = 25-x2 The graph of the region is  To evaluate: The area…

Q: Find the area of region enclosed by x = t – t“, y =t – t°, 0 <t < 1 using Green's Theorem.

A: The solution are next step..

Q: Find the area enclosed by the given curves. y =x, y= 4x O A. 16 O B. 2 O C. 8 O D. 4

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Q: Evaluate the integral and interpret it as the area of a region. 3x + 28 - x dx (2591 – 224v7 –…

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Q: Evaluate the double integral. / 7(16 + y²) ¯ dA; R. R is the region in the first quadrant enclosed…

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Q: Sketch the region enclosed by 2y = 5/T, y = 3, and 2y +2x 7. Decide whether to integrate with…

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Q: d. Find the surface area of the region R bounded by z = x² + y? + 1, z = 5

A: Use surface area concept

Q: Calculate the integral/ / (x + y)* sin° (x – y) dA, where D is the region given by D = {(x, y) : |+…

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Q: Which of the following represents the area of region B? O9 (2) - f(2) dz Sa f (x) da So f (z) da Off…

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Q: Use an iterated integral to find the area of the region. y = 64 – x2 dy dx = y 60 50 40 30 20 10

A: From the given figure, Now find the x-intercept and y-intercept; x-intercept:…

Q: For the area bounded by y=sin2x + 2 in Quadrant I, with the x axis from x = 0 to x = 2n. Find the x…

A: Given: y=sin2x+2

Q: Find the total area of the region enclosed by y=x^2 and y=2x by integrating with respect to y. A 2/5…

A: topic - area bounded by curves

Q: The double integral ff 2xycosydA over the rectangular region R = {(x, y) E R?, 0< x<1 and 0 <y<n}is…

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Q: Find the total area of the region enclosed by y=x^2 and y=2x by integrating with respect to y. A…

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Q: Sketch the region enclosed by 2y = 5/a, y = 4, and 2y + 2æ = 7. Decide whether to integrate with…

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Q: Use an iterated integral to find the area of the region. dy dx %D 47 y 10 9. 8. 7 4 3F 2 -1 4 -1L

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Q: Which of the following represents the area of the region bounded by y = 2, x = 0, y = 0, z = 0 and y…

A: Use the given information.

Q: Consider the following. y = x2 y = 12 - x 16 14 12 10 y 8. 6. 2 4 6 8 10 12 (a) Find the area of the…

A: Recall: Area , A=∫x=ax=bf(x)-g(x) dx ; where f(x)≥g(x) Area,A =∫y=cy=df(y)-g(y) dy ; where…

Q: (IV) The area of the region R bounded by y = x² and y = Vx, Sfe dA equals : dxdy (b) Sxy dydx (c)…

A: Given: Area of region bounded by y=x2 y=x

Q: 10. Determine the surface area of the part of z = xy that lies in the cylinder given by x² + y² = 1.

A: Note:- Since you have asked multiple question As per our policy, we will solve the first question…

Q: 1. Let A be the region enclosed by the curves y = r² + 1, y = (2x – 1)² and r = 1. 2 ty y = x² +1…

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Q: Evaluate f S Sp(2+ e(=²+v²+z*)*/²) dV, where B = {(x, y, z) | x² + y² + z² < 9} Describe the region…

A: Consider the provided question, (2) (a)

Q: Sketch the region enclosed by 2y = 5/T, y = 3, and 2y+2x = 7. Decide whether to integrate with…

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Q: Evaluate the double integral / | x(1+y²)¬1/² dA; R is the region in the first JR quadrant enclosed…

A: Given data: The given integral is ∫∫Rx1+y2-1/2dA. The first function given is y=x2. The second…

Q: By sketching the graph and using integration, find the area of the region bounded by the ellipse and…

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Q: Given the functions: 1. y = 0.1sin((x+0.15)/-0.1) +0.1 2. 6y=x^2-10 Find the area of the region…

A: Let R be the region bounded by the curves y=0.1sinx+0.15-0.1+0.1and 6y=x2-10 the region is shown in…

Q: 3) Evaluate fſ x²dA; for the region in the first quadrant enclosed by xy = 1, y = x, y = 2x.

A: The given double integral is I=∫∫x2 dA To find the limits of integration we will plot the given…

Q: Lalculate the integral | (x+ y)² sin´ (x – y) dA, where D is the region given by D= {(x, y): |2| +…

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Q: Evaluate the integral R(x² – 2y²) dA, where R is the first quadrant region between the circles of…

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Q: Find the area of region enclosed by x = t – t", y =t – t°, 0 < t < 1 using Green's Theorem.

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Q: Find the area of the upper portion of the cylinder x² + z? = 1 that lies between the planes x = ±1/2…

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Q: Determine by direct integration the second moment of area about the x-axis [units^4] of the area…

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Q: Consider the following. y = x2 y = 12 - x 16 14 12 10 y 10 12 (a) Find the area of the region by…

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Q: Find the area of region enclosed by x = t – tª, y = t – t2, 0 <t < 1using Green's - Theorem.

A: Given      x=t-t4   ,  y=t-t2   ,   0≤t≤1

Q: Find a limit equal to the area shown. y| y= In(x+ 1). 1 3 2. 2.

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Q: Find the total area of the region enclosed by y=x^2 and y=2x by integrating with respect to y. (A)…

A: To find the area of the region enclosed by the two curves.

Q: Determine the area of the region between the two curves in the given figure by integrating over the…

A: We will find the area between the two curves as following

Q: The double integral S xdA over the rectangular region R = {(x,y) E R², 0 <x< 2 and 0 < y< 1} is…

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Q: Which of the following shaded regions has an area equal to In 2? y = 1/x y = 1/r A. 1 3/2 2 2 В. 4 y…

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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.