Find the area under the curve for the parametric function defined by the equations x(t) = t³ +t + 1, y(t) = t² – t, and 1

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

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Q: Evaluate the unoriented line integral (22 + y° + z²) ds (2x² along the parametric curve a : [0, 1]…

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Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: One-form ω=xydx+12ydy and the parametric curve α:0, ln2→ℝ2 defined by α(t)=e2t, e-2t. To…

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

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Q: Calculate the integral of f(x, y, z) = 3x² + 3y² + zª over the curve c(t) = (cos t, sin t, t) for 0…

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Q: 3. Evaluate the line integral f. F· dr where F = (ev+ye")i+(xe"+e")j and C is the parameterized…

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Q: Find the area of the closed region enclosed by the curve defined parametrically by x = 3 cos t, y= 2…

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Q: Find the area under the parametric curve x=t(1+t) , y=1/(1+t) ,  between t= [0,1]

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Q: compute the line integral f(x, y) ds for the given function and path or curve. f(x, y) = x – y, the…

A: Given: fx,y=x-y To Find: Line integral ∫Cfx,yds over the unit semi-circle x2+y2=1, y≥0

Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: Evaluate the line integral / V(ry + xz + yz) · dĩ over the curve F(t) = (t, 2t, 3t) for 0 <t < 4…

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Q: Evaluate the line integral for the given curve C. S-ds, C: r(t)= (t',t*),1<t<2 cy ||

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Q: Find the area bounded by the parametric curve x=t−(1/t), y=t+(1\t) and the line y=2.5.

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Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve.  xy2…

A: Our Aim is to evaluate the line integral along the given positively oriented curve xy2 dx+3x2y dy…

Q: Let f(x,y) = x 3 + y 4  + sin(2x+3y) . Determine the line integral of f(x,y) with respect to arc…

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Q: Evaluate the integral [(e' - y)dx + (sin y +x)dy using Green's theorem where C is 16 bounded by y =…

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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 a scalar line integral to find the length of the curve r(t) = (20 sin , 20 cos ,5), for 0 < t <…

A: Given

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: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t > 0. COS So r(t)dt Compute the integral above.

A: ∫0πr(t)dt=∫0π&lt;t,3cos t,3sin t&gt;dt               =∫0πtdt i+∫0π3cos tdt j+∫0π3sintdt k…

Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: ω=xydx+12ydy To find: I=∫αω  where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…

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

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Q: Find an integral for the area of the surface generated by revolving the parametric curve about the…

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Q: Evaluate the integral |xy dx – z dy + 8z dz where C is the curve with parametric equations x = 2t, y…

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

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Q: Consider the curve described by the parametric equations x = t° +1, y= t² – º. Compute the area…

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Q: Calculate the integral I = | (2a – y) dx - (a – y) dy where L is the curve represented as x = a(t –…

A: (2a - y) = 2a - a(1 - cost) = a(1 + cost) (a - y) = a - a(1 - cost) = acost dx = a(1 - cost)dt dy =…

Q: Given the scalar function f(x,y)=2xy,calculate the line integral along the ellipse curve from point…

A: Given data:- The given curve is, x29+y24=1 The given scaler function is, f(x,y)=2xy t E 0,π/2

Q: Consider the line integral xy dx + xy° dy with C the triangle with vertices (0, 0). (1,0), and (1,…

A: Given, the line integral Where C is the triangle with vertices.

Q: Let w = e" dx + 4x dy + sin(x1³) dz be a one-form on R³. Find the line integral of w going…

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Q: Evaluate the unoriented line integral (2æ² + y² + z²) ds along the parametric curve a : [0, 1] → R³…

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Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. Find a vector parametric equation for the parabola y = x²…

A: The solution is following.

Q: | 5y? dx +xy dy

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Q: Integrate ƒ(x, y, z) =-2x2 + z2 over the circle r(t) = (a cos t)j + (a sin t)k, )  0&lt;=…

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Q: Evaluate the line integral F. dr, where F (x, y) = (3+2xy) i+ (x² – 3y²) j and C is the curve given…

A: Given problem:-

Q: Find the area under the curve for the parametric function defined by the equations x(t) = 5t, y(t) =…

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Q: Find the area under the curve for the parametric function defined by the equations x(t) = -2 cos t,…

A: Given: xt=-2cost  , yt=3sint , 0≤t≤π2 Formula used: The area under a parametric curve: A=∫abytx'tdt

Q: Evaluate the line integral Sp(xi+ yyj) · dr where D is the oriented curve shown below. (1,1) y = x²…

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

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Q: Find the area enclosed by the given parametric curve and the x-axis. x = 4 sin(t), y = sin(t)…

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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: Evaluate the unoriented line integral +y² + z²) ds along the parametric curve a : [0, 1] → R³ with…

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Q: Evaluate the line integral: f y² dx + x² dy by parameterizing C where the curve C is a square R with…

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

Q: Evaluate the integral S 4-y2 dx dy using polar coordinate

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Q: 1. The area under the curve with parametric equations z = 1 + t?, y = l(2 – t) in the interval 0sts1…

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Q: Find the area under the parametric curve x= 2(t + cos t), y=2(1 - sin t) from t=0 to t= pi.

A: We have to find the area under the parametric curves whose equations are : x=2t+cos ty=21-sin t…

Q: Find the coordinates of the centroid of the curve x = cos t + t sin t, y = sin t - t cos t, 0…

A: Given,

Q: Compute the line integral [(yze²v² + y² + 2 − 2x) dx + (xze²v² + 2xy) dy + (xye²³² +x+32²) dz, where…

A: We have to compute the line integral ∫Cyzexyz+y2+z−2xdx+xzexyz+2xydy+xyexyz+x+3z2dz, where C is the…

Q: The area of the surface obtained by rotating the curve y = et +e asxsb, about the x-axis is 1/2 e2 +…

A: Find the surface area rotating the curve about x axis

Q: Evaluate the integral: L x2 + y² dy dx by using polar coordinates.

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