The area of the region between the curve y = xr? – 6x +8 and the x-axis & r = 0 ; x = 3 is given by - 6x+8)dx +| (x — бх + 8)dx а. b. [(x – 6x + 8)dx 3 |(x² – 6x +8)dy с. 3 d. - бх +8)dy + [(x - 6х +8)dy

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

Q: fo axy dx + 2xy dy, where Cis the rectangle bounded by x = -1.x = 1, y = 1, and y = 5. 6xy dx + 2xy…

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Q: Use the Midpoint Rule with the given value of  n toapproximate the integral. Round the answer to…

A: Width of each interval = (4-1)/6= 3/6=0.5  So the partition is 1, 1.5, 2, 2.5, 3 , 3.5 , 4 The…

Q: Which of the following integrals represents the length of the curve y = 2*, 0 < x < 3? 3 VI+ 2* dx…

A: Given function :

Q: Calculate (x²+y²) dx dy area when D is the limited to x²-y²=1, x²-y²= 9, xy = 2, xy=4

A: As per the question we are given a double integral over the 2d region D bounded by the following…

Q: ven f(x,y) = -2xy² where x = 5t and y = 8t2. Findf' (t) at t = 3. %3D

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Q: Evaluate the integral (2x – y) dx + (x + 3y) dy along the path C. C: x-axis from (0, 0) to (0, -3)…

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Q: 4. Find the volume of the solid generated by the curve f(x)=4-cos.x revolved 360° about the x- axis…

A: Please check the region, given region is not bounded, we consider the region bounded by curve, x=0,…

Q: Solve for the areas of each color using Definite Integration. A (,V 25-x² dx) – (L, V16 – x² de) =…

A: Given:

Q: D. Use double integration to find the area of the region R enclosed by the parabola y = 4-x² and the…

A: Let's find area by using double integration.

Q: (a) The Composite Trapezoidal Rule applied to the integral - [ f(z)dx I = gives the error b - a, 4h³…

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Q: .5 Approximate J, In x dx using the Midpoint Rule with n = 4. Round your answer to five decimal…

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Q: Evaluate the line integral (-3x+y)dx + (x + 3y)dy, where C lies along the x-axis from x = 0 to x = 6

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Q: The vahe of the integral fx*dx with step length 0.5 using Trapezoidal's nule is a) 0.375 b) 0.875 c)…

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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: Solve dy / dx = 1 /( x^2 – 1) , for y(0) = 0. Find its maximal interval of existence %3D

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Q: y = x², y = 6x, about the line y = 36 can be computed using the method of disks or washers via an…

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Q: dx Evaluate J, +6x+10 by dividing the interval into 8 equal parts by using Trapezoidal rule.

A: The integral ∫15dxx2+6x+10. We have to evaluate this integral by dividing the interval into 8 equal…

Q: Q2 Using forward and backward difference approximations of O(h) and centered difference d²y…

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Q: Ⓒ Given SS² +8 dy dx. Reverse the order of in tegration and evaluate. Draw the region.

A: In the given problem first we draw the region then reverse the order of integration and after that…

Q: Use either integration by substitution or integration by parts as needed to evaluate S³= A B x²+1…

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Q: Sin²y dy V1 -x, given that y(-3) = 0. dx 2) Snow that the general solution for (x2 +x-6)dy = 4y dx…

A: Given that  The following differential equations

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.

Q: The volume of a right circular cone of radius x and height y is given by V=1/3πx2y. Suppose that the…

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Q: Determine whether the following statements is true or false, and explain why.

A: Answer: False.

Q: Integrate f(x,y) = x-5y over the curve C: x +y = 64 in the first quadrant from (0,8) to (V32,/32).…

A: Formula Used:Line integral ∫Cf(x, y) ds=∫abf(x(t), y(t)) dxdt2+dydt2dt

Q: c) Evaluate Ssin 3xcos 4x dx by using integration by part.

A: Given integral is: I=∫sin3xcos4xdx Simplifying it:…

Q: Q1) Find only the limits of the integration of the figure below in the order (dx dy). 1-S[ dxdy x=…

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Q: 3. Find the area of the region that lies inside both curves: r = sin 0,r = cos 0 0.8 0.6 0.4 0.2…

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Q: Find the real number a for which the value of the iterated integral V1- (y- (-2²- ỷ +2y) dx dy a-1 -…

A: Let fx,y=-x2-y2+2y. To determine the maximum value of the function, differentiate f(x,y) partially…

Q: y? = 1? Which of the following integrals represents the length of the upper half of the ellipse 25…

A: Topic = Application of integration  Correct answer = A

Q: 1. a- Find a result by using variable separation for dy dx b- y³y+x³=0

A: Note:- These are multiple questions and acording to our company guideline we can do maximum 1…

Q: Hn obfect Is moving along the X- axis with acceloration givon by at) =Q +42zt. TF V l0) =1 and…

A: Given query is to find the displacement function for the acceleration given.

Q: hich of the following is not variable separable? (А) 2ху dx + (х2 + 1) dy %3D0 A ye' dy - (e-y+…

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Q: 4-x2 xe?y dy d.r, 4 - y Sketch the region of integration C, reverse the order of integration, and…

A: First I have see the limit of integration for x and y and then draw the region and then taking strip…

Q: V = 200 /R volts in free space in the region bounded by 2mm <R <3mm, 0 <8 <n / 2 and 0 <o <n / 2…

A: Given 2 mm&lt;R&lt;3 mm, 0&lt;θ&lt;π2 and 0&lt;ϕ&lt;π2 Now, 2×10−3 m&lt;R&lt;3×10−3 m And under…

Q: +5y )dx+(2x-4y) dy edary of the region lying between

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Q: Which of the following integrals represents the length of the upper half of the ellipse = 1? 16 3…

A: we have to find the length of upper half of the ellipse

Q: Sketch the region whose area is represented by the definite integral. ▪4 5 dx y y 6F

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Q: c) Evaluate Ssin 3.xcos 4x dx by using integration by part.

A: Here we have to find            ∫ sin 3x cos 4x dx Integration by parts formula is          ∫ uvdx =…

Q: 2 -πx²y. = The volume of a right circular cone of radius ï and height y is given by V 3 dy when x 4…

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Q: Ql: f(x.y)-y cos xy over the rectangle 0sxs , 0 sys1

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Q: a²f, a2f ax ay2 ду? f(x, y) = In(x2 + y?) + tan-1() A- Show that = 0, where B- Evaluate the integral…

A: Dear students you have asked multiple questions. As per our policy your first question will be…

Q: Find the area between the curves y=e−0.2xy=e-0.2x and y=1.6x+1y=1.6x+1 from x=0x=0 to x=4x=4.

A: Given query is to find the area under the curve

Q: The value of (4xy- (sinx)³)dx + (x² + y9)dy (Correct to ONE decimal place) where C is the region…

A: Let F=Px,yi+Qx,yj be the vector field and C be the boundary of the region D. To evaluate the given…

Q: The dimensions x and y of an object are related to its volume V by the formula V=9x2y How is dV/dt…

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Q: If f=(3x?-3yz)i +(3y² – 3xz)j+(3z² – 3xy )K, then the value of the integral (f.dī along the curve x2…

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Q: Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's rule to approximate the integral 2/3 | (3…

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Q: Evaluate [ sin (3x +3y)dA where D is the region between x +y' 1 and x+y 7.

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Q: Q5: find the possible solutin area: Max Z= 8x+ 6Y S.To 2X+ 2Y ≤60 2X +4Y ≤ 48 X,Y20

A: To find the solution we have to plot the constraints. Here we use the graphing tools to find the…

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.