Let D be the region in the first octant bounded by the coordinate planes, the plane r+z = 1 and the cylinder r = 1-y². Which of the integrals below are equivalent to the integral //e²+y°+s° dV ? D (1-y I. + dz dr dy Jo 1-z L[ * dy dr dz 0. IL * III. dr dy dz + 1-y2 * dz dy dz (a) I only (b) II only (c) III only (d) I, II (e) I, III

Q: 7. Determine the surface area of the solid generated by /9 – r² where –4 < IS

A: Area of surface of rotation about x-axis is given as, A=∫ab2πy1+dydx2dx Here, given that y=9-x2. So,…

Q: S how that the line af intesection of the Planes and 5x-2y-Z=0 X+2y-22= 5 is Paralkel to thelime メ+3…

A:

Q: Find the area bounded by the region y = 2x + 2 and the lines y = 1, y = 2

A:

Q: 6. Find the area of the part of the surface z = x² - y², which lies in the first octant that is…

A: The given problem is to find the area of part of the surface, which lies in the first octant that is…

Q: 2. A solid is in the first octant lies inside the hemisphere z = 4 - x² - y², outside the cone z =…

A:

Q: 9. Find the area of ze zy that lies inside the cylinder z² + y² = 1.

A: In this question, we will find the area of the surface z=xy that lies inside the cylinder.

Q: Evaluate the double integral  SSDxydA where D is the triangular region with the vertices (0,0)…

A: ∬DxydA, where D is triangular region with vertices (0,0), (3,0) and (0,1).

Q: 3) Evaluate the double integral fS, dr dy where D is the region bounded by the hyperbolas ry = 2, ry…

A:

Q: Evaluate yºda – x³dy, where C' is the boundary of the region between the circles œ² + 1² = 1 and x²…

A: The given problem is to evaluate the given line integral over the given path C where C is boundary…

Q: 7. −ydx + (x + √xy)dy = 0

A:

Q: Find the area of the shaded region. 22) y = (x 3)2 (2,1) (4,1) 4. C)- D) Evaluate the integral. 31/4…

A: Other

Q: 2. Calculate the centroid of the thin plate R which bounded by r = 1, y = 0 and, 2² + y² = 4.

A:

Q: 9. Find the volume of the solid in the first octant enclosed by the surface z = x? and the four…

A: Volume = ∫02∫03∫0x2dx dy dz                 =∫02dx∫03dy ∫0x2dz…

Q: Let W be the region bounded by z = 9-y.y = 3x², and the plane z = 0. (a) Calculate the volume of the…

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.