The volume of the solid generated when the region at the first quadrant bounded by the curve y = x² + 1, the y-axis, and the line y = 5 is revolved about the line y = 5.

Q: Find the volume of the se

A: Given, the curve                   y2(a+x)=x2(a-x)

Q: The closed curve x²+y²-14x-14y+49=0 revolves about the line x=-3. Determine the volume generated.

A: Given the closed curve is x2+y2-14x-14y+49=0 and the line x=-3

Q: Find the centroid of the region cut from the first quadrant by the circle x2 + y2 = a2.

A:

Q: Find the area of the propeller-shaped region enclosed by the curve x - y3 = 0 and the line x - y =…

A: x-y3=0  &  x-y=0⇒y3=x  &  y=x⇒y=x13  &  y=x Now the diagrammatic representation of the…

Q: Calculate the area of the region that lies between the line y=-1 and the parabola y = x²-6x-1,

A:

Q: find the parabolas x= y2,y= 255¢ and the ine area Of the region bounded by the

A:

Q: The centroid of the area bounded by the parabola y^2 = 4ax and the line x = p coincides with the…

A: Please see the attachment

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.