The curve y = (64x – 4x³ )m bounds the shaded section on top and the x-axis on the bottom. Determine the area of the shaded section and the coordinates of its centroid.

Q: Determine the centroid of a cross section with an isosceles right triangle of base 9 cm, a…

A: Determine the centroid of a cross section with an isosceles right triangle of base 9 cm, a…

Q: Find the average squared distance between the origin and thepoints on the paraboloid z = 4 - x2 -…

A: Given: Paraboloid: z=4-x2-y2, z≥0 To find the Average squared distance.

Q: Find a center of mass of a thin plate of density 8 = 3 bounded by the lines y = x and x = 0 and the…

A:

Q: integrate the function f(x,y) = x^2 +y^2 over the unit circle

A: In this question we  integrate the function f(x,y)=x2+y2 over the unit circle the unit circle…

Q: find a parametrization for the curve. the upper half of the parabola  x-9=y^2

A: The given curve is x-9 = y2  ...(a) Putting y = t in (a) then x = 9+ y2…

Q: Find the area of the cap cut from the paraboloid y^2+z^2=2x by the plane x=2

A: To Find: Area of the cap cut from the paraboloid y2+z2=2x by the plane x=2.

Q: Use the Disk Method Y= x^(3/2) , Y = (3/2)x , revolved around X = 5

A: First we have to design the integral using the given conditions,  and for that obtain functions of…

Q: Find the centroid of the region bounded by the curves y=2x-4, y=2x1/2, and x=1. Sketch the graph.

A:

Q: Find a center of mass of a thin plate of density 8 = 5 bounded by the lines y = x and x = 0 and the…

A: Center of mass of the plate is given by  x,y=MyM,MxM here M=∫∫D δx,ydAMx=∫∫D yδx,ydAMy=∫∫D xδx,ydA…

Q: Find the area of the surface when y = vx, from (4, 2) to (9, 3) is rotated about the x-axis.

A: Surface area formula for rotating about x-axis is S=∫2πydS where dS=1+dydx2dx

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.