(c) (i) By using polar coordinate, evaluate ff, x2 + y² dA, if R is the region in between circle x? + y? = 1 and x? + y? = 4 in the first octant. (ii) By using double integrals, find the area of the regions enclosed by y = sin x ,y = cos x ,x = ,X = 4

Q: 2. Find the surface area obtained by rotating the curve (a) y = sin(Tr) around the x-axis for 0 <x <…

A: The objective here is to find the volume of area obtained by the rotating the given curves in each…

Q: Evaluate the tripple integral of f(x,y,z) = sin(x2+y2)  over the solid cylinder with height 5 and…

A:

Q: 3. Set up an integral to find the surface area generated when the given curve x = where 0<t < 1 is…

A:

Q: Set up, but do not evaluate, an integral for the length of the curve. y = 6 cos x, Osxs 27 dx

A:

Q: Given [[(3xy)dA, where R is the region bounded by y=0 and y = /9-x² . Sketch R the region of…

A:

Q: |-4 Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. I. $. (x…

A: According to the given information it is required to evaluate the integral:

Q: A6] Evaluate the integral J: J over the area of a circle of radius 2, using polar coordinates.

A: We have, J=∫∫Ax2y2dxdy   Equation of circle is, x2+y2=22   In polar coordinates, x=rcosθy=rsinθ Here…

Q: 4. Find the volume of the solid generated by the curve f(x)=4-cos.x revolved 360° about the x- axis…

A: Please check the region, given region is not bounded, we consider the region bounded by curve, x=0,…

Q: 7. For some microphones, the region of sensitivity takes on the shape of a rotated cardioid Consider…

A:

Q: Calculate the integral (x – y) dx + (x + y) dy I = x2 + y2 where L is along the curve y = cos x,…

A:

Q: 8. By using the double integral in polar coordinate, find SS e-x*+y*)dx]dy

A: As we know that the polar coordinate: x=rcosθ,y=rsinθdxdy=rdrdθ Given: ∫0∞∫0∞e-x2+y2dxdy So, the…

Q: Which of the following integrals will find the area of the surface generated by revolving the curve…

A: Given Data: Function: f(x)=4cos(x) Interval: a≤x≤b=0≤x≤π3 The derivative of the given function with…

Q: Sketch the region bounded by the curves y=e* , y=0, x=-1 and x=1. ...

A: The region bounded by the curves is shown in the figure below

Q: 1/ Calculus

A:

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

A:

Q: Q3/ Evaluate: 1- SS (4x + 2) for the region enclosed by the curves y = x² and y = 2x 2- Show that…

A: 1. Consider a function of two variables, z=fx,y. The double integral of function fx,y is denoted by…

Q: Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve…

A:

Q: Evaluate the given integral by changing to polar coordinates. x² - y² dA, where D is the region…

A:

Q: Evaluate the double integral I. y sin(a* )dA R where Ris the triangular region with the three…

A:

Q: valuate the given integral by changing to polar coordinates. || sin(x² + y2) dA, where R is the…

A:

Q: Use Green's Theorem to evaluate the line integral. (e-x²12-v) dx + (0-1²/2 + x) ay dy C: boundary of…

A:

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: Set up an integral for the area of the surface generated by revolving the given curve about the…

A:

Q: Use a double integral in polar coordinates to find the region described: The region in the first…

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Q: Consider the following curve. y = x³, osxS 5 Set up an integral in terms of x that can be used to…

A: The surface area of revolution about the x-axis is: S=2π∫abf(x)1+f'(x)2dx If the function is,…

Q: Given ||(3xy)dA, where R is the region bounded by y=0 and y = 19-x? . Sketch R the region of…

A: We will find out the required value.

Q: Set up an integral for the area of the surface generated by revolving the curve y = x2 about the…

A:

Q: 1 dA , where R lies in the second quadrant and is bounded by the graphs of y = -x, the y-axis, and…

A: Use  ,  and    to convert Cartesian form to polar form . Note : The area can represented by .

Q: 10. Use a definite integral to find the exact area enclosed by the x-axis and the parametric curve =…

A: The given parametric curve is: x=2t3+1    and     y=2t-t2

Q: 3. (a) Find the volume of the solid generated when the region enclosed by the curves x = Jy and x =…

A: Given that  x = sqrt(y) x = y/2

Q: 2. The curve is given as a curve with positive direction in motion at the boundaries of the square…

A: As per Bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: у 3 1 + sin х, 0 <x< т/2 y

A: The given curve is y=1+sinx and the limit is 0≤x≤π2 The Surface area is given by S=∫x2πy1+y'2dx or…

Q: The graph above shows how the x-axis, the y-axis, the line y = -x + 3 and the curve %3D y = 2 sin x…

A:

Q: Evaluate the line integral fe 3y dx + z? dy + (25 - x) dz over the quarter of the circle of radius 5…

A:

Q: to polar coordinates turns into an integral S S" x² + y² dxdy When converting the integral

A:

Q: Set up and evaluate the definite integral for the area of the surface generated by revolving the…

A: y=16-x2,  -1≤x≤1 Differentiating with respect to x, we get…

Q: Which ONE of the following integrals correctly represents the area of the surface of revolution…

A: giveny=ln(x5)rotation about x-axisA=∫152πyds

Q: 6. By using the double integral in polar coordinate, find ff e-(x²+y')dA for the region enclosed by…

A:

Q: 3. Set up an integral to find the surface area generated when the given curve x = where 0 <t < 1 is…

A: x=t3,y=t2      o≤t≤1    Given

Q: T Express the surface area of the surface obtained by rotating the graph of y = 4 sin (x) for 0 ≤ x…

A: Recall: Surface area SA=∫ab2πy1+dydx2dx We have, y=4sin(x), a=0, b=π6

Q: 7. Evaluate the integral I | sin(9.r² + 4y²)dA by hand, using an appropriate change of variables,…

A: The given question can be solved as shown in step2

Q: A) If the volume (1) of the region (D) enclosed by the surfaces (z = x² + 3y²), and (z = 8 – x? –…

A: Answer A:   The intersection between surfaces: x2+3y2 = 8 - x2 - y2 2x2+4y2 = 8 x2+ 2y2 = 4 (x/2)2…

Q: which of the following integrals represents the area of the surface obtained by rotating the curve…

A:

Q: 1. Consider the double integral // sin(g) dA, where R is the region in the first quadrant of the…

A:

Q: By using double integrals, find the surface area of plane + y² -+z=c that lies inside a a the…

A:

Q: Set up an integral that represents the area of the surface obtained by rotating the curve = e2t, y =…

A:

Q: - SxS Write an integral for the area of the surface generated by revolving the curve y = cos (3x)…

A:

Q: Evaluate the integral: L x2 + y² dy dx by using polar coordinates.

A:

Q: Apply Green’s Theorem to evaluate the integrals ∮(3y dx + 2x dy)C: The boundary of 0≤ x ≤π, 0 ≤ y ≤…

A: According to Green's theorem - ∮Pdx+Qdy = ∬D dQdx-dPdydxdy          ...............(1)Given integral…

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.