24. Find a parametrization of the lemniscate (x2 + y²)² = xy (see Figure 23) by using t = y/x as a parameter (sce Exercise 23). Then use Eq. (6) to find the area of one loop of the lemniscate. y 0.5 -0.5 0.5 -0.5 FIGURE 23 Lemniscate.

Q: 1. Prove that fo r*dr + rydy = where C1 is the line segment from (0,0) to (1,0) by using…

A:

Q: F D' Point D is the Nof AABC because AF, BG, and CE are all centroid incenter Submit Answer…

A: We know that, The circumcenter of a triangle is the point where the perpendicular bisectors of the…

Q: 8.Evaluate the triple Integral. Te by The paraboloid X=2yt + 2z? and the plane X=2. 7x dV, where E…

A:

Q: Evaluate the spherical coordinate integral. 1- cos 2 2x x SS S p²sin dp do de 00 1- cos 2 2x x SS S…

A:

Q: Evaluate the line integral [(x+ y)dx+(x - y)dy, where C is the arc of the parabola y = 2x from (0,…

A: The integral in the form ∫CM(x,y)dx+N(x,y)dy involves the integration over a specific path C. In…

Q: Which of the following(s) is/are applicable to solve 3e dx – 6xe dy over triangle whose vertices are…

A: To find method applicable to solve ∫C3e-ydx-6xe-ydy over triangle whose vertices are (0,0),(4,-1)…

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: Z=fGx,y)= within Regron %3D メ -४ Sketch and label region (ラ×30 Set up rals SS f(x, y) dA as SS S…

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: 18. Show that the centroid of the sector in Figure 13 has y-coordinate 2R sin o (0, y) R FIGURE 13

A:

Q: 1 Find the area of the surface obtained by rotating the curve y = In(1 – x²), 0 < x <. about the

A:

Q: 1. Let C be a path along the upper half of a circle from the point P(2,0) to the point Q(-2,0).…

A:

Q: 6 foND THE EXACT LENGTH Of THE zt -2t PARAMETRIL y(e)=1+4t CURVE X= e te

A:

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

A:

Q: Calculate the line integral ∫Cy2dx+xydy, where the curve C is the part of the ellipse (Figure 6)…

A: To calculate line integral  where C is an part of ellipse parameterized by where

Q: 4. Evaluate the line integral 6. x²y dx – xy² dy where C consists of the arc of the circle 16 = x² +…

A: Given equation of circle is  x2+y2=16x2+y2=42 radius is 4 We have to calculate line integral ∮ cx2y…

Q: Calculate the volue of the de integral 23 - 1 2 F soro fallowing c a) Cil2+1|= 1 Curves. 6)…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 12. Consider the lamina covering the regon x442-2,0EXET WH aunsity functionplx,u) = - 5x + 2u+5. 1ts…

A:

Q: Give the integral representing the length of the parametric curve x=t^3, y=t^4, 0≤t≤1.

A: Given: A parametric curve:   Now the length of the parametric curve is given by the formula:

Q: 2. Evaluate directly the line integral fo a dy where C is the quarter-circle from (0, 4) to (4, 0)…

A:

Q: 14. (a) Give a parametrization for the intersection of y = x² and z = e". (b) Set up an integral…

A:

Q: 6. What is an equation for the largest ball centred at (0,0, 1) that can fit in the paraboloid z…

A:

Q: Compute the irertia móment by x ( Ix=Sy?JA) tor number off segment n=s hyte multple-trapezoidal…

A:

Q: Find the Nork do ne against (x,Z)ih muving a particle 11 along the curve parametrized

A:

Q: Evaluate ſ[ſp z dV where E lies between the spheres x² + y² + z² = 1 and + y*¨+ 2² = 4 in the first…

A: Solution:

Q: Calculate the area under y = x2 over [0, 1] using Eq. (11) with the parametrizations (t3, t6) and…

A:

Q: Compute where Cis the counter clockwise oriented boundary of upper-half unit disc D. 0.66666666667…

A: We need to find ,  ∮ y2dx + 3xydy , over  C ,  where C is the counter clockwise oriented boundary of…

Q: .... ........ 4.8. Find the area of the surface generated by revolving the curve x = cost, y =…

A: See the details solution in below

Q: Describe the bifurcation for the following dy - Ay - y %3D dt

A: The saddle node-bifurcation: Let y˙=f(α,t) is the system of equation, the to find the saddle-node…

Q: 45. If A = (y – 2x )1 + (3x + 2y)j, compute the circulation of A about a circle C in the xy plane…

A:

Q: At what points does the curve r(t) = ti + (3t - t2)k intersect the paraboloid z = x2 + y2? (If an…

A: Given- r(t)=ti^+(3t-t2)k^z=x2+y2 To find the intersection point of the given curve and paraboloid.

Q: 5/10 Determine the coordinates of the centroid of the shaded area. y * = ky? a b.

A: Given:

Q: 2.7 Describe cll soilutions of Asso where A is the Yow ta the matirix meg Vse Y S, and/or t as in…

A: Given  A=1-6-50-2-100100-200001-7000000 Here, we have to solve Ax=0

Q: | 5y? dx +xy dy

A:

Q: Find The or rthoganal tajeclolies. ! d faly Curver r= 4C cose, where C be being parameer

A:

Q: Use the separated solutions to solve the Laplace equation. in the region 0 0 given the boundary…

A: The Laplace Equation Uxx+Uyy=0with(a) U(0,y)=U(1,y)=0(b) U=0 as y tends ∞c) U(x,0)=sin5(πx)

Q: Sketch the bifurcation diagram for y = (A– 4t)y (where A is a parameter, as usual). 2

A: To find- Sketch the bifurcation diagram for y' = λ - 4t2y (where λ is a parameter, as usual)

Q: Given the scalar functionf (x, y) = 2xy, calculate the line integral along the ellipse curve ²² +²²=…

A: We have to find the line integral along the given curve.

Q: Calculate the line integral f(-102 + 4z + 5) de + (-7z – Oy? – by – 4) dy, where Dis the annulus…

A:

Q: Consider the following. C: line segment from (0, 0) to (4, 8) (a) Find a parametrization of the path…

A:

Q: 1. Draw a sketch of the cylinder z= ==V36 – 4y2.

A: Given: z=1336-4y2

Q: # 4. Use Green's Theorem to evaluate the line integral P (y*dx +2xy° dy), where C is the positively…

A: Given integral is, ∮Cy4dx+2xy3dy

Q: In which of the choices below is a parametrization for the tangent line to the curve of intersection…

A:

Q: 3 2 Use a line integral to calculate the length of the helix ř(t) = (a cost, a sint, bt) 0<t < 2n…

A:

Q: Compute the area A under the parametrized curve c(t) = (e',5t) for 0 <t < 1. %3D (Give an exact…

A:

Q: Suppose F(2, y) = -y sin(æ)i + cos(2)j. (a) Find a vector parametric equation for the parabola y =…

A: Given F→(x,y) = -ysin(x)i→+cos(x)j→ a) Let us assume x=t then y=x2=t2 by differentiating r→(t) with…

Q: (9) Evaluate by converting to cylindrical polar coordinates 322dV where E is the solid bounded by y…

A:

Q: 1. Find parametrizations of the curve of intersection of x² + 4x + z² = 0 and x- – 2y + z = 3.

A: Given curve is x2 + 4x + z2 = 0 .  This can be written as (x+2)2 + z2  = 4  So the curve is a circle…

Q: Find a parametrization of the curve x^(2/3)+y^(2/3)=8^(2/3) and use it to compute the area of the…

A: Finding the parameters:

Q: b2 Let c lbe twisted eubic Curine parametrizes by for osts% flaxt367) ds. Find

A: Let C be twisted cubic curve parametrized by γt=ti+t2j+t3k for 0≤t≤12. We have to find ∫C8x+36zds.