Evaluate C, dxdy, by first reversing the order. . Se ye" x' i) Draw the region R for the above problem.

Q: 2) Evaluate JJ (x – 1)dA; for the region in the first quadrant enclosed by y = | x³, y = x.

A: Given that, ∬(x-1) dA ; for the region in the first quadrant enclosed by y = x3,  y = x. First, draw…

Q: Find the area of the region enclosed by one loop of the curve  r= 3cos 40

A: We have to find the area by one loop of the curve r=3cos4θ We know area A=12∫-aar2dθ for cosθ we…

Q: 5. Find the area of the region enclosed by one loop of the curve: r= 4cos(3Ø)

A:

Q: et l2 be the same region as in part (2). Write this region in the form 12 = {(z, y) : e<ysd: a(y)…

A:

Q: H.w./Find the area forthe fottowing regtont 4)Region inside the circle ((=2sine) on left of the line…

A:

Q: c) Find the length of the boundary of region R.

A:

Q: Compute I| (x – y) dA, D is the region in the second quadrant enclosed by x2 + y2 = 5 and x2 + y? =…

A:

Q: Evaluate JJ xy dA, where R is the region in (1, 3) (5, 3) (2, 1) (4, 1)

A:

Q: 3. Evaluate ff V²+ y²dA, where R is the region bounded by the circles with center (0,0) and radius 3…

A: Since you have asked multiple questions, we will solve the first question for you. If you want to…

Q: Find the aroa of shaded region in the given figure (2. 4) y- 2N O 14 O 413 7/8

A: We have to solve given problem:

Q: If Dis a simply connected, closed region in the plane and the boundary of Dis defined by the…

A: If D is a simply connected closed region in the plane and the boundary of D is defined by the…

Q: 3) Find the area of the region enclosed by the cardioid r=1-cos®

A: See the details solution in below

Q: Give area of the region enclosed by y and y = 0 from x = 1 to x = 3. 2 A. 2 В. 3.4 С. 1.5 D. 1

A:

Q: Find the area of the region bound by the following curves: y = /x, y= 6 – x, y = 1. O 17/6 O 1/2 O…

A:

Q: 10. Find the area of the region bounded by y = =.y = 1, and r = 3.

A:

Q: Consider the following shaded region. YA y=ax - x? y=bx Find the area S of this region if a = 4, b =…

A:

Q: Find the area of the region inside the inner loop of the limaçon r= 1 + 2 cos 0.

A:

Q: Given the region bounded by the graphs below, find the area of the plane region. y=3x+3 and…

A: First recognise the region bounded and  find where they intersect each other. Accordingly find the…

Q: Consider the following shaded region. y=ax – x? y=bx Find the area S of this region if a = 4, b = 2.…

A: Firstly we will find the point of intersection from graph as shown below and then find area of…

Q: Evaluate xy dA, where S is the region shown in Figure 16. (2, 4) 2 Figure 16

A: Let us consider the double integral ∫S∫f(x,y)dxdy, ∫s∫f(x,y)dydx,  To solve the double integral…

Q: 3) Find the area of the region enclosed by the following line and the curve. y = x – 1 and y² = 4x +…

A: The graph for the given functions y=x-1 and y2=4x+1 is shown below.

Q: 11. Find the area of the region enclosed by x+y? = 0 and r+3y? = 2.

A:

Q: Q1/ Sketch the region defined by <0< sos , -1 <r<1 Ans./

A: 1. Consider the given information. -π4≤θ≤π4, and -1≤r≤1

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

A:

Q: 6- Region shared betweenthe cadioids (r=1+ coso)and (r=l-Caso).

A:

Q: Sketch the region given by the set. {(x,y) | xy &gt; 0}

A: Given: To Sketch the region given by the set{(x,y)|xy&gt;0}

Q: 49. SR 3xy dA; R is the region in the first quadrant bounded by y = 2 x, y = 0, and x = 4 − y². =

A: Given Region  is bounded by  curve y=2-x, y=0 , and x=4-y2  in first quadrant

Q: If R is the region enclosed by x- axis, y = -3x and x = -2, then evaluate the following double…

A: Let us consider a function z=f(x,y) then the double integral over the region R is denoted as…

Q: O Evaluate (9te"")dn + (ex+ Gary)dy where C is the boundary of the region enclared by 4x* and x=y?…

A:

Q: (2) Sketch the following region R. R= {(r, 0) : 1 < r< 3, 7/2 < 0 < 27} %3D

A: R=(r,θ): 1≤r≤3, π2≤θ≤2π Since 1≤r≤3 and π2≤θ≤2π

Q: 3. Evaluate sse- Ne-cx)) JA where D isthe region shown below ニxく ソュ -x

A: The given integral is,∫∫D(3-6xy) dA

Q: 2. Find the area of the indiconted region. Between y=x3 and y=1 For x in El,ņ I for x in El,]

A:

Q: 5) Find the area of the region R that lies inside the cardioid r= 2(1+ cos 0) and outside the…

A: To find the area of region R that lies inside the cardioid r=2(1+cos θ) and outside the circle r=3.

Q: 22 Evaluate f, e-(x²+y?) dx dy, where R is the region x? + y? < a?.

A:

Q: as Find Hhe voleime of odid generated by revolving tthe region bounded by graphs of y=xe;y 0, and…

A:

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

A: We first find where they meet

Q: 2. Find the area of the region enclosed by æ + y? = 0 and ¤+3y² = 2.

A:

Q: 10- Evaluate JS where E is the region that lies inside the cylinder x² + y²dV E x? + y? = 16 and…

A:

Q: of the region en and y =x? +4

A: 2. Given curve is: y=-x2-2x When y = 0 we get, -x2-2x=0⇒xx+2=0⇒x=0 ; x=-2 Then the area is:…

Q: Find the area between the loops of the limacon r=2(1+2cosθ).

A:

Q: 4. Evaluate f, 2ry dA, where R is the region bounded by y i and

A:

Q: Q4: Evaluate , xy²dA over the region R 中 shown in the accompanying figure. -3 -1.5 1.5 3

A:

Q: Q4: Evaluate S, xy²dA over the region R Y shown in the accompanying figure. X -3 -1,5 1.5 3

A: We have to evaluate the integral over the given region R:  ∫∫R xy2 dA

Q: A sketch of the region enclosed by y = x and y = f Area of the region = fot 4.25 2²+1 follows.

A: Given: y=x and y=4.25xx2+1 First find the intersection point Thus, x= 4.25x1+x2 ⇒1+x2= 4.25⇒x=…

Q: as given by shaded region in Figure. R

A: The double integral is given as ∫∫Rx2+y2dxdy . Convert the given integral in polar coordinates ,…

Q: Let D be the region bounded by (a) y = (0 < a< 1),

A: As many parts and its further sub parts are given, so I will solve only one ( Question (a) and its…

Q: Graph the given curve and find the area of the region enclosed by one loop of the curve. r = 4 sin…

A:

Q: Sketch the region enclosed by ?=5?y=5x and ?=3?2y=3x2. Decide whether to integrate with respect to…

A: Given: y=5x, y=3x^2

Q: Compute the area of the shaded region. y -x+8 -8 128 4. O 16 64

A:

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.