Which integral computes the circumference of the ellipse + y° = 1? %3| 2 16 — За? dx 16 — 4г2 2 -2 2 16 — 4г2 da 2

Q: x³ +1 Q3: Find the length of y=√x+2,1 ≤ x ≤ 7, Using ds = 1+²dy into the surface area of the object…

A: Find the arc length of y=x+2, 1≤x≤7 using the differential element ds=1+dxdy2dy. For formula for…

Q: solve part (d) Consider the ellipse +5 = 1. (a) Show that the circumference of the ellipse is given…

A: Suppose u=gx and g' be a continuous function on a,b and also f is  continuous over range of the…

Q: Find the area enclosed by the ellipse x = a cos t, y = b sin t, 0 ≤ t ≤ 2π.

A: Given Data The first equation of ellipse is x=a cos t. The second equation of ellipse is y=b sin t.…

Q: find the area of the surface S: x² + y² + 2² = 4 -A <tan -^ (*) FO 6 < COS Cos FJ VI (2) 4 DEX TI 3…

A: The given Surface is                    x2+y2+z2=4 ........1where π6≤tan-1yx≤π3 and π4≤cos-1z2≤3π4…

Q: Calculate the value of x in the figure: AB = BC = DA and BD = DC (Not drawn to scale). %3D %3D 21 2x…

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Q: Evaluate the integral / sin (9x? + 4y?)dA, where R is the region in the first quadrant R bounded by…

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Q: te the line integral c 3 == 1 from A=(2, 0) to (0, -1) Jax + 2xyay where is the arc on the ellipse

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Q: Consider the integral dx I = 3 x+2 The approximation of I using the two-point Gaussian quadrature…

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Q: Q3 Point out the singularity and is ty pe of follo wing fune tions 22 () d) Sin t2 24

A: Singularity

Q: Use Y=sin) and reductiori of order to find the generale soletion to the homogeneous differenticl…

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Q: Find Df and Duf to:- XZ on the point (,,1) inthe direction of V = 2i -3j+6 K. %3D soLiin ansi- lo

A: Directional derivative of the function.

Q: Compute sin z dz , where a is the piece of the parabola with equation y= a?, which lies between the…

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Q: Given the path C that is the line segment from (-4; 0) to the origin, and the curve y = 2x² from the…

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Q: Find the surface area of the figure generated by revolving x = 6 cos(t), y = 6 sin(t) where 0 <ts…

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Q: y = x², X = about the line x = -1 can be computed using %3D the method of disks or washers via an…

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Q: Find the length of the spiral c(t) = (t cos t, t sin t) for 0 <t < 2n to three decimal places. Hint:…

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Q: Find the anea of the vegion in quadrant Ps bou nodod ist 4 the conie the by the right by the line…

A: Given curves, y=ex+12,y=e-x+12,x=2 ln2 To find the area of region bounded by these curves.

Q: Q3(a) Solve the integral below by using polar coor dinate and sketch the region 2-yt 2xy dxdy J-/2-y

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Q: Using Green’s formula, evaluate the integral ∮C(x+y)dx −(x−y)dy, where the curve C is the ellipse…

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Q: +y²)dx + (x² – y²)dy where C is the triangle with vertices (0,0), (2,1) and (0,1).

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Q: Q.1) Use the polar integral to evaluate 2 dydr 71+ Vr2 + y?

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Q: Q3) find the soluation of differention equation 1) 2y' = y cot(x)

A: To find the solution of the differential equation 2y'=y cot(x) Formulae used : ∫dxx=ln |x|∫cot(x)=ln…

Q: dx Evaluate LJv by using Trapezoidal rule (up to three 1 + x? decimal places). [Ch-4(B), Expl-2]

A: We have to evaluate ∫01 dx1+x2 by using Trapezoidal rule.

Q: Which of the following is the surface area of the surface generated by revolving the curve y = 폴 in…

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Q: Calculate the integral | (y? + 2?)dx + (2? + a² )dy + (x² + y²)dz where C is the curve intercepted…

A: C is the intersection of two surfaces : x-r2+y2=r2z2=2R-xx The coordinates will be :…

Q: Evaluate o y dx – x³ dy, where C includes the two circles of radius 2 and radius 1 centered at the…

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Q: 3) Evalnete the itevated integral by Converting the to polar corrdinate, a) dy dx

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Q: Q1/Find the distance between(5,3I10to(5, Z10) in Cylindrical Coordinnde? Ans:lo

A: According to question Distance between 5,3π2,0 to 5,π2,10Let P1x1,y1,z1 and P2(x2,…

Q: au +20 dy 10 www ax dy dy?

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Q: s being pumped into a spherical ballo nge of the radius when the radius reac

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Q: Given the area bounded by the parabola x² – 4x + 12y- 20 = 0, the x- axis and the lines x = 1 and x…

A: Sol

Q: A circular cylinder of radius 1 and the parabolic cylinder x = 2y? – 1 2 =

A: In this question, we have to find the curve of intersection.

Q: Find the area enclosed by the ellipse x = a cos t, y = b sin t, 0&lt;= t&lt;= 2pai.

A: Given,

Q: х 3D0, у%3D1, = y° about the line y = 1 can be computed using the method of disks or washers via an…

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Q: 7.38. Calculate the second moment of area for a 2 × 6 piece of lumber about the x-x and y-y axes, as…

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Q: O is the centre. EO = /3 . OF = 2 and OE I DF. Calculate the length of DE. (b) %3D F. E V3 2]

A: From the given problem:   EO=3 ;OF=2  OE⊥DF.   As we know that perpendicular from the center of the…

Q: 4. Find the value of the integral 223 + 5 'zp- c (z – 1)(2² + 9)* taken counterclockwise around the…

A: The objective is to find the value of the integral ∫C2z3+5z-1z2+9dz

Q: Find the area enclosed by the given ellipse. x- a cos t, y = b sin t, 0sts 2r

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Q: Show that the arclength of the graph of y = sin x on the interval [0, 27] is equal to the circum-…

A: The length of the smooth curve ζ : [ 0 , 2π] →ℝ2  is given by I(ξ) = ∫02π ξ'(t)  dt

Q: WXYZ is an isosceles trapezoid with bases WZ and XY and midsegment MN. Use the given information to…

A: Given  WXYZ is a isosceles trapezoid with  MN= 2x+1 XY=8 WZ=3x-3

Q: 7. Evaluate the integral I | sin(9.r² + 4y²)dA by hand, using an appropriate change of variables,…

A: The given question can be solved as shown in step2

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

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Q: In the figure to the​ right, the equation of the solid curve is y equals StartFraction secant…

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Q: *Q. ( 12 ), p. 410 : Find the area of the surface generated by revolving eУ + е-У the curve x = 0…

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

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Q: What is the general solution of ydx (* + Vy? - t²) dy %3D = cy arcsin In y = C %3D O arctan In y = C…

A: Convert this equation into the homogeneous differential equation.

Q: Q.01: Use Green's theorem to evaluate the line integral £ e2*sin2ydx + (e²*cos2y)dy, where C is the…

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Q: I = Examine the given integral dz ;C z-i -|= 2 the circle

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Q: Consider the ellipse + = 1. n/2 Show that the circumference of the ellipse is given by C 2 V4 cos? t…

A: According to our company's guidelines I can only answer first three subparts of the question since…

Q: Given the area bounded by the parabola x– 4x + 12y - 20 = 0, the x-axis and the lines x = 1 and x =…

A: ∫ (y^2)dx is the formula for moment of inertia

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.