3. Find the exact area of the surface resulting from rotating y = v1 + e* from x = 0 to x = 1 about the X-ахis.

Q: The area of the surface obtained by revolving the curve y = x3 for 1<x<8 about the y-axis, is given…

A: Surface area obtaining by x=f(y) from x=a to x=b is about y-axis given in step 2.

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

A:

Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

Q: 1. Calculate the integral 1 y-stnx)dx + cosy dy a) With directly (classical), b) With the Green's…

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

Q: (a) Set up an integral for the length of the arc of the hyperbola xy = 1 from the point (1,1)  to…

A:

Q: 3.4.8 Evaluate the area of the surface z = 2-x² - y² lying above the (x, y) plane. (Hint: Use polar…

A:

Q: QUESTION 8 Find the area of the cylindrical surface in Xyz-space where a point (X,y,z) on the…

A: Find the surface area of the cylinder in xyz-space where a point x, y, z on the surface satisfies…

Q: Evaluate the integral / (12x + 3y)dæ + (3x – y)dy along the parabolic arc y = x² from (0, 0) to (1,…

A:

Q: Determine a region of the xy-plane for which the DE (1+ y) y' = x²will have a unique solution whose…

A: Theorem of existence of uniqueness: If f(x,y) and dfdy are continuous on rectangular region R, then…

Q: Suppose S is the surface of revolution obtained by rotating the curve y about the y-axis. = cosx…

A:

Q: The volume of the solid generated by the curve y = e, between z = 2 and z = 6 about the z- azis is

A: Given: The curve is y=e-x   To determine: The volume of the solid generated by the curve y=e-x…

Q: 16 The area under the curve 4y? (x2 + 1)3 was rotated about a line located to the left 9 of y – axis…

A:

Q: 16. Compute the surface area of the surface obtained by revolving the given curve about the…

A: To compute the surface area of the surface obtained by revolving the given curve about the axis x =…

Q: 17. Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1x 4, 0 y < 1

A:

Q: The surface area of the surface generated by revolving y= 1 – x² for 0<x<1 about the x – axis is s=…

A: The solution is as follows

Q: 2. Use Green's theorem in order to compute the line integral P (x – 4)³ dy – y³ dx where is the…

A: We need to compute the given integral using Green's theorem . ∮C(x-4)3dy-y3dx Where is the circle of…

Q: 4. Evaluate the line integral 6. x²y dx – xy² dy where C consists of the arc of the circle 16 = x² +…

A: Given equation of circle is  x2+y2=16x2+y2=42 radius is 4 We have to calculate line integral ∮ cx2y…

Q: 2. Show that the surface area of the solid of revolution obtained by revolving the curve y = 1/x, x…

A:

Q: Evaluate the integral (12x + 15y)dx + (24x – y)dy along the paraboli arc y = x from (0,0) to (1,1).…

A:

Q: .... ........ 4.8. Find the area of the surface generated by revolving the curve x = cost, y =…

A: See the details solution in below

Q: By using polar coordinates, evaluate the double integral -y²dx dy x² + y² over the region in the…

A: Answer is given using polar coordinates.

Q: # 4. Evaluate the line integral | (x +2x²y) ds, where C is the upper half of the unit circle x2 + y2…

A:

Q: Set up the integral for the Surface Area of the curve y = In(x) from (1, 0) to (e, 1) %3D rotated…

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: The shape of a redwood tree (without any branches) can be modeled by an upside down paraboloid, f(x,…

A:

Q: Evaluate the line integral (2 - u) dz + (3 + æ) dy] where C is the parabolic arc = a² from (0, 0) to…

A: We have to evaluate the line integral.

Q: 3. By using Green's Theorem in the plane, evaluate | (y - sinx) dx + cos x dy where C is the…

A:

Q: Use Green's Theorem to evaluate the line integral. y dx + 7x dy C: square with vertices (0, 0), (0,…

A:

Q: 2. Evaluate O (2y + In(5x² +3)) dx + (10x 3) dy along the boundary of the elliptic- region 4. y? 1…

A: Introduction: Green's theorem is very efficient in calculating line integral. When the line intgeral…

Q: Sketch the regions enclosed by the given curves. y = 4 cos(7x), y = 4 sin(14x), x = 0, x = T/14 y y…

A:

Q: 2.7 Describe cll soilutions of Asso where A is the Yow ta the matirix meg Vse Y S, and/or t as in…

A: Given  A=1-6-50-2-100100-200001-7000000 Here, we have to solve Ax=0

Q: The surface area of the solid of revolution generated by rotating x = (y² + 2)²,1 < y s 2 about the…

A:

Q: Evaluate the integral (6x +12y)dx + (6æ — y)dy along the parabolic arc y = x² from (0,0) to (1,1).…

A: Let us evaluate the given integral in the next steps.

Q: The surface area of the surface generated by revolving y= 1-x2 for 0<x<1 about the x - axis is S = […

A:

Q: 5. Compute the volume of the solid obtained by revolving the curve y = vx sin x about the x-axis…

A:

Q: 6. Use Green's Theorem to evaluate the line integral fc 2xy dx + (x² + y²) dy, if C is the ellipse…

A: Let R be simply connected region with a piecewise  smooth boundary  'C'. If functions 'M' and 'N'…

Q: 2 Sphere S to be the top half of sphere · x² + y² + 2² SS₁zdS /5 calculate the integral.

A: The given Surface is the top half of the surface x2+y2+z2=a2. To Evaluate: ∬zdS.

Q: Compute the integral |(5y + pz2020, ) dx + (19x – e-4ycos(v) sin y) dy over the portion of the…

A:

Q: (b) Find the area of the surface obtained by rotating the region bounded by the curves 7 35xy 5y12+…

A: To find the area of the surface obtained by rotating the region bounded by the below curves about…

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: Evaluate the triple integral III xy dV where E is the solid tetrahedon with vertices (0, 0, 0),…

A: The given integral is, ∫∫∫E xydV Here E is the solid tetrahedron with vertices O0,0,0, A10,0,0,…

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: #3. Evaluate the line integral O (y dx- x dy) where C is the positively oriented circle with center…

A:

Q: Q2// Find the resulting volume of the curve y = sin x cos x rotation about the x-axis if x=0, x=/?

A:

Q: 1 9. Evaluate dV;where E is the solid between the spheres of radius 1 (x² + y² + z²)³/2 E and 2…

A:

Q: Evaluate the integral / (12x + 18y)dx + (12x – y)dy along the paraboli - arc y = x² from (0,0) to…

A:

Q: Use Green's Theorem to evaluate the line integral along the gi | 4y3 dx - 4x³ dy Cis the circle x2 +…

A:

Q: 3. Set up the formulation to find the surface generated when the curve y = x² is revolved about the…

A: 1st we need to convert the curve in x=f(y) form and corresponding limits for y , Then use given…

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.