3. Suppose the curve x = t³ – 9t, y = t+3 for 1

Q: 3. Set up an integral to find the surface area generated when the given curve x = where 0<t < 1 is…

A:

Q: Show that the curve r= Vti + Vtj + (2t – 1)k is tangent to the surface x +y² – z = 1 when t=1

A: Given that r=ti+tj+2t-1k and f=x2+y2-z-1 Take the derivative for f, ∇f=ddx,ddy,ddzf=2x,2y,-1…

Q: 7. Let C be the curve y = 3x for 0 <x S 1. 0.5 Find the surface area of revolution of C about the…

A: Given: y=3x3 for 0≤x≤1

Q: |-4 Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. I. $. (x…

A: According to the given information it is required to evaluate the integral:

Q: 5. Evaluate the integral xdx+ (x+y)dy where C is the curve x 1+y from (2,1) to (2,-1).

A:

Q: Evaluate the line integral y = x² from (-1, 1) to (1, 1) and the line segment from (1, 1) to (−1,…

A:

Q: x = 7 te', y = (t + 1)e', 0stsi Set up an integral that represents the area of the surface obtained…

A:

Q: Suppose S is the surface of revolution obtained by rotating the curve y about the y-axis. = cosx…

A:

Q: 10. Evaluate the integral cz(z-1) where C is a simple closed curve in the positive direction that…

A:

Q: Evaluate the line integral [21r²yzds with respect to s along the 2 curve C : x = t₁ y = t², z =…

A:

Q: Consider the curve defined by the equation xy=25. Set up an integral to find the length of curve…

A:

Q: = x (with y > 0) about the y-axis can be computed using the method of disks or washers via an…

A: Given: Let us first plot the region.

Q: 16. Compute the surface area of the surface obtained by revolving the given curve about the…

A: To compute the surface area of the surface obtained by revolving the given curve about the axis x =…

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 3y3 dx…

A:

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 5y +…

A: Use Green Theorem to evaluate  ∫C5y+9exdx+10x+7cosy2dy where, C is the boundary of the region…

Q: =t°, y = t² 3. Set up an integral to find the surface area generated when the given curve x = where…

A: Given:      x=t3, y=t2 Interval 0≤t≤1   Surface area generated about x-axis is…

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 4y dx -…

A:

Q: 5) Evaluate the integral (e* - y) dx + (sin y + x)dy using Green's theorem where Cis bounded by y =…

A:

Q: Evaluate the following line integral by two methods: (a) directly; (b) using Green's Theorem. $y²dx…

A:

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve 5y³ dx -…

A: We know that , Green's theorem ,  Consider C be a positively oriented  , simple , closed , piecewise…

Q: Consider the curve segments: 1 - to x = 4 and 4 1 X = -to x = 16. 16 S1: у %3D = x² from x = S2: y =…

A: Given that:

Q: The area of the surface obtained by rotating the curve y = 1 I+ X, 1 <xse about the line x = 28 is…

A:

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

A:

Q: Suppose C is the curve rt) = (71.2³), for 0sts4, and F= (6xy). Evaluate F Tds using the following…

A:

Q: 2. Evaluate fy³ dx-r³ dy, where C is the circle of radius 2 centered at the origin, in the following…

A:

Q: Set up an integral to find the surface area generated when the given curve x = t3,y = t2 where 0 ≤t…

A: The curve is given as , x=t3 , y=t2  where 0≤t≤1 find surface area when curve rotated about the…

Q: Set up an integral to compute the arc length L of the curve C: r (t) = (t², 4t, v/2t) from the point…

A: The given curve is: r(t)=12t2, 4t, 2t From point (0, 0, 0) to point 12, 4, 2 the limit of t is:…

Q: Set up an integral to compute the arc length L of the curve C: r (t) = (t, 4t, V2t) %3D from the…

A: It can be found by integrating between limits of t= 0 to 1

Q: Use Green's theorem to evaluate the line integral along the given positively oriented curve. [(5y +…

A:

Q: Evaluate the line integral Jo 9x²yzds with respect to s along the с 2 curve C: x=t, y = t², z = /t³…

A:

Q: 1+y dy

A: We can solve the given integral as follows:

Q: Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise. y10 dx + x6…

A:

Q: Compute the integral „2020 (5y + x*0 ") dx + (19x – e -e-4yCos(y) sin y) dy over the portion of the…

A:

Q: Calculate the integral of C being the curve given below. y= 3+x y= 3-x 6x dS C

A:

Q: Consider the following curve. y = x³, osxS 5 Set up an integral in terms of x that can be used to…

A: The surface area of revolution about the x-axis is: S=2π∫abf(x)1+f'(x)2dx If the function is,…

Q: Evaluate the line integral ∫Cxdy−ydx along the curve C defined by the equation y=x3 from the origin…

A:

Q: Consider the following. x = t − 2 sin(t), y = 1 − 2 cos(t), 0 ≤ t ≤ 8? Set up an integral that…

A: Our objective is to setup thr integral

Q: Set up an integral to compute the arc length L of the curve C: r(t) = (, 4t, v2t) %3D from the point…

A: I am going to setup an integral to compute the arc length of the given curve using the required…

Q: Use Green's theorem to evaluate the line integral along the given positively oriented curve. | 6y3…

A: given circle x2+y2=4

Q: Evaluate the line integral, where C is the given curve. |(2y3 - Vx)dy, c is the arc of the curve y =…

A:

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 7y3 dx…

A: See attached file for a step by step explanation to evaluate the given line integral.

Q: The area of the surface obtained by rotating the curve y = 1 < x s e about the line x = 24 is O 2n…

A:

Q: Express the area of the surface obtained by rotating the curve y =In sec(2x) between x = 0 and x =…

A:

Q: 1 (#3) Find the surface area for y = 3 on [0, 3] rotating on the x- axis.

A: Given an equation of a curve, The solid will form by rotating the curve about the x-axis.

Q: 3. Set up an integral to find the surface area generated when the given curve x = where 0 <t < 1 is…

A: x=t3,y=t2      o≤t≤1    Given

Q: Use Green's theorem to evaluate the line integral along the given positively oriented curve. | 8y dx…

A: Introduction: Green's theorem is one of calculus' four fundamental theorems, all of which are…

Q: ху — 4, 1 <x< 8

A: Surface area.

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 2y3 dx…

A:

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

A:

Q: Set up and evaluate the definite integral for the area of surface generated by revolving the curve…

A:

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.