Consider a thin metal plate that covers the triangular region shown below. If the density in the plate is measured by 8(x,y) = 7x²y³ -2x'y then calculate the plate's mass using the formula: mass= 8(x,y)dx dy where R= ((x,y): 0 s xs1&0sys 2} 1. %3D R y 2.

Q: y = 4 − x2,    y = 0 Calculate its area

A:

Q: a) Enter a definite integral that gives the volume of the solid when the region above is revolved…

A:

Q: Ris the region bounded by the functions f(x) = 2 + Jz and the g(x) and the lines x=0 and x=4.…

A:

Q: (a) / #(8+ 2²) dz = +C. -3x6 + 6x12 dx = (b) +C. -3 (c) / (6 + x*)² dx = +C.

A: Explanation of the answer is as follows

Q: Calculate the volume of the solid of revolution (or show why it does not exist) obtained by rotating…

A: We have to calculate volume of solid of revolution obtaining by the rotating the function.

Q: Use cylindrical shells to find the volume of the solid formed by rotating the area between the graph…

A:

Q: D. Calculate volume enclosed by (z = 8-x² - y²) and (z = x² + y²), (Use order dy dz dx). وكنت طول

A: Given

Q: Answer the given DE below: x (dy/dx) +y=y^2

A:

Q: | sec 5x dx

A:

Q: A solid is formed by rotating the region bounded by the curve y = e-5/2 and the x-axis between a = 0…

A: Since the solid has a constant density So the centre of the mass is equal to the centroid.. So we…

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.