8) Find the area of the region between the curve x = e- e15t Le-14t and the x-axis, 0

Q: Let R be the region bounded by the curve y = r and the line y = 2x. Use inequal- ities to describe R…

A: In this question we have to describe the area of region by using vertical cross section

Q: Sketch the region R whose area is given by the double integral. 3 19 - y2 dx dy

A:

Q: Evaluate the double integral. (5x – 7y) dA, D is bounded by the circle with center the origin and…

A:

Q: af 11. Suppose + = 0, prove that the line integral | F. fe -dx + ду dr = -dy is inde- pendent of…

A: Here F = fy i+fxj

Q: Evaluate the line integral [6. 6ayⓇ ds, where C is the right half of the circle x² + y² = 4.

A: To evaluate :   ∫C6xy6ds, where C is the right half of the circle x2+y2=4 Prerequisite : We can…

Q: Compute the line integral (Баллов: 2) Jo ds, Cis the right half of circle x2 + y 16 (counter…

A:

Q: Compute the area of surface generated by revolving the four-cusped of hypocycloid x⅔+ y⅔ =a⅔ about…

A:

Q: Compute the surface area of revolution about the ?-axis over the interval [0,?] for ? = 6sin(?).

A:

Q: The area bounded by the curve z 4y+7, the y-ax y =0 and y 2 is dy3D O 18 22 11 O 15

A: Area bounded by the curve can be calculated as

Q: find the ora of the Region described t vegion bounded by gee"iy=e EIny and

A:

Q: Ques tion No. 02: Show that the area of f egion R encleaseod Ly simple closed cuve č is given by a…

A: Area of closed curve, using Green's Theorem: A=∯Sdxdy   =12∯S2dxdy   =12∯S∂x∂x+∂y∂ydxdy…

Q: Evaluate the line integral xdy- xydx over the closed path C where Cis the circle C

A:

Q: Use Green’s Theorem to evaluate  ∫ sin 5y dx + 5x dy  around the boundary curve C of the region R,…

A:

Q: Use Green's Theorem to evaluate the line integral f(e*c - 2y)dx+(5x+ev* dy, where C is the 15. 2cos:…

A: We need to find integral.

Q: Find the area of the surface generated by eY + e -y revolving the curve x = in the interval -In 2…

A:

Q: Use Green's theorem to evaluate O(1-y') dx+(x³ +e" ) dy , where C is the boundary of the region…

A: The integral is ∮C1-y3dx+x3+ey2dy where C is the boundary of the region between the circles x2+y2=4…

Q: 5. Find the volume of liquid needed to fill a sphere of radius R to height h (Figure 18). R h…

A:

Q: Use Green's Theorem to calculate line integral o sin(x²) dx + (3x – y) dy where C is a right…

A:

Q: Consider the region enclosed by the curves y = a2 +1 and y = 2r + 1:

A:

Q: Let CR be the circle of radius R centered at the origin. Use Green's Theorem to find the value of R…

A: ∮ y3 dx+x dyUsing green's   theorem∮P dx+Q dy=∫R∫∂Q∂x-∂P∂y  dAP=y3                  Q=x∂P∂y=3y2…

Q: 1. Sketch the region enclosed by the given curves and find its area. y = |a| and y = VT+ 2,

A:

Q: 5 Compute the Surface area f obtained by revoluing the line y=x around the y-aris. a cone whichü

A: To find the surface area of cone which is obtained by revolving y=x revolved around y-axis.

Q: 5. Evaluate the double integral (2x – y) dA where D is bounded by the circle with the origin as its…

A:

Q: Calculate the area of the intersection of the regions limited by curves r = 1 and r = 2(1 − cosθ).

A:

Q: egral dS over the su 9, where n is th surface element

A:

Q: Sketch the region enclosed by the given curves andfind its area.   y - cos πx    , y=4x2- 1

A: The given curves are y=cosπx   and  y=4x2-1. Sketch the region enclosed by the above curves as shown…

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

A:

Q: 3. Evaluate ff Vr² + y²dA, where R is the region bounded by the circles with center R (0,0) and…

A: Given integral is: ∫∫Rx2+y2dA Where R is the region bounded by the circles with center 0,0 and…

Q: Consider the region bounded by the Xxy curves J 3-2x, and y=3, %3D

A: Calculating the area of the region bounded by the curves

Q: ae e drea under the curvé y=*from2-2to2 using a Rightt Endpoint approximation with 6 subdivisions..

A:

Q: The area bounded by the curve y = In(z) and the lines z = 1 and =2 above the z -azis is %3D square…

A:

Q: 2. By integration, show that the area of a circle of radius r is A = rr?

A: According to our guidelines, we can answer only one question or three subparts of a question. Other…

Q: Find the area of the surface generated by revolving the curve x in the interval 2 Osys In6 about the…

A:

Q: The area enclosed between the curves p(a) =2 +*+ 3 and g(@) = -22 - * + is O 16 47 %3D 18 49 O -49…

A:

Q: Evaluate the line integral f. y³ dx – x³dy, where C is the positively oriented circle of radius 2…

A:

Q: Evaluate the line integral fx ds between (4,0) and (0,4), where C is the circle of radius 4 centred…

A:

Q: It is possible for a region in the xy-plane to be both type 1 and type 2 at the same time.

A: This is possible.

Q: Find the area of the surface generated by eY + e -Y revolving the curve x = in the interval 2 In 5…

A: solution:-given curve x=ey+e-y2 in the interval 0≤y≤ln5revolving about the y-axisarea of a surface…

Q: Find the area enclosed by the closed curve obtained by the joining the spiral r=4θ by a straight…

A: Given that, the value of r=4θ and θ∈0,1.7. It is known that, the formula for the area of the…

Q: Compute the line integral Sy?a dl where y is the part of the circle centered at the origin and of…

A:

Q: y = sec 0 tan 0 →0 TT 4 4 -Vz

A: We have given a diagram;

Q: (a) Find that the area of the plane region lying to the right of the parabola x = and to the left of…

A:

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

A:

Q: Use Green's Theorem to evaluate the line integral (у — х) dx + (2х — у) dy for the given path. C:…

A:

Q: Sketch the region enclosed by the given curves. y = 4 cos(5x), y = 4 – 4 cos(5x), 0sxS a/5

A:

Q: 23. Find the volume of the solid formed by the region in the first quadrant bounded by: x = y-y3, x…

A: Consider the graph:

Q: 8K Use Green theorem to eva luate the . line integral : 3. 3 where c is ニ thecircle

A: Given the line integral

Q: 8. Find the area of the surface of the region generated by rotating the curve y = x³ from x=0 to x=1…

A:

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

A:

Q: Sketch the region enclosed by the given curves. y = 8 cos(x), y = 8 - 8 cos(x), 0<x T

A: This question is related to area bounded by curves, We will use integration to solve it.

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.