Use the formula A = 27 dy 1 + dx dx to determine the surface area of the parabolic reflector obtained by ro- tating the curve y = Vx, 0 < x < 1, about the x-axis.

Q: the surface area of the surface generated by revolving the curve c(t) = (t, sin(t)) about the x-axis…

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Q: For the curve y = f(x) = x², 0 < x < 2, its length is 2 dy` dx - {(aV17+ In (4 + v17). VI7). L 1+…

A: Apologies for the inconvenience. The correct solution is given below: Given that y=x2 Hence, dydx=2x

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A: Intregation of product of two variables is given below

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Q: Plot the following quantities versus distance along the beam a) Displacement (y). b) Slope 0(x) =…

A: Note: Since you have posted multiple sub parts. As per our policy we are allowed to solve the first…

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Q: Find dy/dx and d^2y/dx^2 X=t-e^t, y= t+e^-t For which values of t. Is the curve concave upward?

A: Given that, x=t-e^t, y= t+e^-t

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Q: Find the approximate value for f(r, y) = sin(a7zy + In(y)) at (0.03, 1.04). Take = 3.

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Q: Given x 13t - 12 and y = 169 sin t %3D %3D dy %3D dx dy dæ?

A: Given that       x = 13t - 12   and   y = 169 sin t Here we have to calculate     dy/dx         and…

Q: For the Parametric curve x = a(t – sint), y = a(1 – cost) where 'a' dy at t = dx is a constant. Find

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Q: Find dy/dx and dy/dx. x = e°, y = te-t dy %3D dx %3D For which values of t is the curve concave…

A: Given: x=ety=te-t Find dydx and d2ydx2 as shown below.

Q: Consider the curve parametrized by x www = 1 + ln(t) and y = t¹ + 2t. dy a) First eliminate the…

A: topic - derivatives

Q: The area of the surface obtained by rotating the curve y = 1<xse about the line x = 32 is ** 12 (32…

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Q: Given the equation x In(y) dz = z Arctan(3y). Assuming z as a function of variables x and y, find ar

A: To partially differentiate the function z with respect to x.

Q: Find the surface area of the object obtained by rotating y = sin(2x) , 0<x< about the x-axis.

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Q: Find the length of the are of the curve п x=e' sint , y=e' cost from t-0 to t=:

A: Given: x=etsint, y=et cost from t=0 to t=π2

Q: For the curve y = f(x) = x², 0 ≤ x ≤ 2, its length is 2 L = 1+ dy dx dx =

A: Given curve y=f(x)=x2,0≤x≤2 The objective is to find the length of the curve.

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Q: compute the length of the curve over the given interval.  r(t) = (cost,sin t,t3/2), 0 ≤ t ≤ 2π

A: Derivative of a function is represented as the change in variable with respect to change in another…

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A: We have to find derivative

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A: We have to find the linear approximation of the given function around suitable point.

Q: fnd he general sotetion ef me dAfferentral equation 6iy ty=2e t40 Sin smx de?

A: The objective is to solve the differential equation 8d2ydx2-6dydx+y=2x+40sinx

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A: Step:-1 Given that x216+y29=1---(1) and x4+y3=1---(2) Using graph, we can see intersecting point of…

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A: Answer is optional B

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 :…

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Q: Find the length of the arc of the curve x=e'sint , y=e'cost from t=0 to t= п b) d) 1

A: To find: The arc length of the curve: Given: The equations are x=et·cos t and y=et·cos t and the…

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A: Sol

Q: Find the area of the surface generated by revolving the curve x = 6t , y = 4t2 for t = 0 to t = 1…

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Q: Find a dy + y dx, where C is the arc of the curve y = e* for 0 < x <7

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Q: Q3- Use Green theorem to evaluate [7y-ex]dx+ [15x-sin(y³ + 8y)] dy Where (c) is a positively…

A: This is a application of integral problem

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Q: Find the lenght of the curve y = x- tnx, 1+/dy 2. dx %3D %3D

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Q: Find x dy + y dx, where C is the arc of the curve y = e² for 0 < x < 1

A: As per guidelines i can solve only first question

Q: ind an equation of the tungent line to y= 45in2 X at point (i II

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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.