3. Explain how we know that it is possible to cut a surface with continuous boundaries in exactly half -- into two pieces with the same area. The answer should include a theorem that proves this.

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: Find the area of the “triangular” region in the xy-plane that is bounded on the right by the…

A:

Q: If a rectangle has its base on the r-axis and two vertices on the curve y = e-", show that the…

A:

Q: Calculate the area of ​​the space limited by the curve y = 4x - x^2 and the axis OX

A: Given: The curve y = 4x -x2  To find: The area of the space limited by the curve y = 4x-x2  and the…

Q: What is the area (in square units) of the region bounded by the graphs of y=x+2, and the x axis from…

A: topic - area under the curve

Q: Find the areas of the regions enclosed by the lines and curves x + y2 = 0 and x + 3y2 = 2

A: Consider the curves ,x+y2=0 and x+3y2=2To calculate the area of the region enclosed by the given…

Q: Sketch the region in the xy-plane defined by the inequalities x- 4y 0, 2- x - 7lyl 20 and find its…

A: Given region are   x-4y2≥0 and 2-x-7|y|≥0

Q: Describe the region R bounded by the curve y = x² and the line y = 2r in terms of inequalities using…

A:

Q: The area of the region enclosed by the curve 4x2+25y²=50 is:

A:

Q: Sketch the region in the xy-plane defined by the inequalities x - 2y2 ≥ 0,   1 - x -| y| ≥ 0, and…

A: Now the graph is,

Q: Show that a closed right circular cylinder of given surface has maximum volume if its height equals…

A: Let, the surface area be S, radius be r, height be h and volume be V. The total surface area be…

Q: Find the area of regions enclosed by curves y x3 – x + 3 and y = x2 + 5x + 3.

A: Given equations of the curve are:                              y=x3-x+3& y = x2+5x+3

Q: Sketch the region bounded by the graphs of y = 9- and y = 6- x, then find its area. 2

A:

Q: Sketch the region in the xy-plane defined by the inequalities x - 5y2 > 0, 2 – x – 9\y| > 0 and find…

A:

Q: What is the area bounded by the curve x² = -9y and the line y + 1 = 0?

A: Given curve is x2=-9y Line is y+1=0 We need to find the area bounded by the curve

Q: Determine whether or not the region bounded by the curves is vertically simple and/or horizontally…

A: Given :               x = y2  ,  x = 2 - y2

Q: The area of region bounded by the parabola y2 = 4x and the line %3D y = 2x – 4 is

A: We have to find the area of the region bounded by the parabola y2=4x and the line y=2x-4 .

Q: What is the area of the enclosed surface bounded by the x-axis, x = 2 and y=x2 ?

A: The area under the curve can be calculated by the integration. The formula for the integration is…

Q: The area bounded by the parabola y = 4x2, y-axis and the lines y = 1, y = 4 is

A:

Q: What is the area of the region enclosed by the curve y=x^2 and the lines y=x+6, x=0 and x=5?

A:

Q: Find the Area of a Surface of Revolution. y= (x3/6)+(1/2x), 1≤x≤2

A: Given y=f(x)=x36+12x,   1≤x≤2

Q: Sketch (clearly label/shade) the region in the xy-plane that is bounded between the graphs of y = 13…

A: Given: fx = y = 13-5x2                              1gx = y = 11x-3…

Q: Find the areas of the regions enclosed by the curves x3 - y = 0 and 3x2 - y = 4

A: The given curves are x3-y=0   and  3x2-y=4. Rewrite the above equations as  y=x3  and  y=3x2-4. Now…

Q: Find the area of the region enclosed between the curves defined by the equations y = x² – 2x + 2 and…

A: Solve the following

Q: 3. Show that a closed right circular cylinder of given surface has maximum volume if its height…

A: Let r, h be the radius and height of the cylinder respectively and V, S be the Volume and Surface…

Q: What is the area of ​​the region between the parabola y=x^2-1 and the line y+2x=2?

A: The solution is given below in the next step:

Q: What is the area of the region bounded by the parabola y^2 = 2x and the line x-y-4 = 0? *

A: See the details solution in below

Q: 4) Calculate the area of the domain bounded by the parabola y = x², the hyperbola y = and the line y…

A:

Q: Sketch the region enclosed by the given curves. 4x y = 1+ x2 4x2 y = 1+ x3 y

A: Given area can be given as below :

Q: 4. Find the radius of the right circular cone with least volume that can be circumscribed about the…

A: Formula Used: Volume of Cone = 13πR2h where R,h denote the radius and height of right circular…

Q: Sketch the region enclosed by the curves y = x2- 4 and y = 3x − 4, and find its area.

A:

Q: Sketch the region enclosed by the curves. x + y = 3, y + 3x = 3, x - y = 0 f(x) = + 10- -8-

A: Drawing the given line.

Q: Describe in words the region in R³ that satisfies 2x - z² ≤ x² + y² ≤ y² + 10

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: Compute for the volume of the solid formed bounded by the equations (y- 4)2 = x - 3, x-10, y-4, and…

A:

Q: In Exercises 25–26, find the areas of the surfaces generated by revolv-ing the curves about the…

A: Given:

Q: Sketch the region in the xy-plane defined by the inequalities x- 5y 2 0, 2 - x - 9lyl 20 and find…

A: Graph of the curves÷

Q: Write down the type of this expression, its surfaces, and plot the region bounded by these surfaces…

A: The equations of two intersecting surfaces are provided as: z2=x2+y2 and x2+y2+z2=2. We need to…

Q: What is the area bounded by the curves y=x2+5 and y=−x2, the line x=3, and the y axis.

A: Given that, The curves are y=x2+5 and y=-x2, the line x=3, and the y-axis. Find the area bounded by…

Q: Find the Area of a Surface of Revolution. y=1/3(x3), [0,3]

A: Given, y=13x3   interval :0,3

Q: Find the area of the region enclosed by parabola y=3-x2 and the line  Y=3-2x

A: We have to find the area of the region enclosed by parabola y=3-x2 and the line y=3-2x. The point of…

Q: Sketch the region bounded by y =x² and x =y ( x 2 0) about the x-axis. %3D

A: Solve the following

Q: 9. Find the demensions of the right circular cylinder of largest volume that can be inscribed in a…

A:

Q: What is the area of ​​the enclosed region between the curves y = x2 and y = 2x - x2 ?

A: We have, the equation of given curves as y=x2                   ⋯⋯1y=2x-x2          ⋯⋯2 On…

Q: Sketch the region enclosed by the given curves. y = 4/x, y = 16x, y = x, x > 0 16

A: We have the following equation   y=4xy=16xy=116xWhere x>0 First, we will make a table for each…

Q: Sketch the region enclosed by the given curves. 1 у %3 2/х, у 3 8х, у %3D —х, 2 x > 0 y 5 4

A:

Q: Find the areas of the regions enclosed by the lines and curves y2 - 4x = 4 and 4x - y = 16

A: Consider a curve: y2-4x=4……1   and line: 4x-y=16……2

Q: Find the areas of the regions enclosed by the curves and lines x = 4 - y2, x = 0

A: The area of the regions enclosed by the given curve x=4-y2 and the line x=0 is shown in the figure…

Q: Sketch the region in the xy  -plane defined by the equation or inequalities. |x|<4 and |y|<2

A: x<4  and  y<2⇒-4<x<4  and  -2<y<2   ∵x<a⇒-a<x<a

Q: Right circular cones of height h and radius r are attached to each end of a right circular cylinder…

A: Let Vcr be the volume of the cone and Vc be the volume of the cylinder. So, Vcr=πr2h and Vc=13πr2h.

Q: y=√x, x = 4, and the x-axis.

A: Consider the given function y=xand the x-axis