Write expressions for the area enclosed by a curve interms of line integrals around .

Q: 5. Sx³e2* dx

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Q: YA y= /(x) 0. C

A: Topic- area under the curve

Q: of unit circle , when F(r)

A: The function given is Fr=-x,-xy here r is the radius of circle ,r=x,y→dr=dx, dy. Apply Green's…

Q: Evaluate the integral using the formulas in the summary and equation as attached

A: Integration is a process of finding the anti derivative of a function. There are basically two types…

Q: Use double integrals to find the area inside the curve r = 5sin(e)

A: The given curve is:

Q: 16-y2 16 – 22-y drdy by using appropriate coordinate system. an

A:

Q: Approximate the area under the curve y

A: Introduction: The formula of the trapezoidal to approximation is given by,…

Q: Find the area between the curves. x = 1, x = 2, y = 3x3 + 2, y = 0

A:

Q: Evaluate the integral using FTC I.

A: Our Aim is to integrate the following expression given below:-∫092 dx-(i)

Q: What change of variables is suggested by an integral containing Vx? + 36?

A: For, an integral containing  x2+36 we can change variable by substituting, x=6tanθ, As it will get…

Q: Find the area bounded by the indicated curves. y = x4 - 18x? + 81, y = 81 - x

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Q: Use six rectangles and right endpoints to estimate the area under the curve f(z) = V2x on [0,3).

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Q: Find the area between the curves. x = 0, x = 6, y = 5x, y = 3x + 10

A: Given: The curves x=0, x=6, y=5x, and y=3x+10 . The bounded area between the given curves is shown…

Q: Evaluate the integrals

A: the given expression is : ∫π3πcot2x6dx

Q: Determine the area enclosed by the curve r²=4sin(20).

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Q: Apply trapezoidal and Simpson's rules to the integral

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Q: Use any of the results in this section to evaluate the given integral along the indicated closed…

A: Cauchy’s Integral Theorem : If f(z) is an analytic function in a simply-connected region R, then…

Q: Write the integral that gives the surface area generated when the curve is revolved about the…

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Q: Find the area bounded by the curve y²=9x and its latus rectum

A: Given equation is y2=9x the equation of parabola is of the form  y2=4axlatus rectum x=a  and…

Q: Find the area under the curve y = 5xe- for x > 4. area =|| %3D

A: Explanation of the solution is given below...

Q: Find the area enclosed by the given curves (using applications of definite integrals).

A:

Q: Write the integral that gives the surface area generated when the curve is revolved about the…

A: Solution: The objective is to write the integral that gives the surface area generated when the…

Q: Evaluate the integral using integration by parts. sin dx

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Q: Evaluate the line integral F.dr

A: In this question, we find the line integral for given vector function and parametric path.

Q: Find the area under the curve y = 3xe for æ > 3. area =

A: Given:The given curve is y=3xe-x and x≥3To find:we have to find the area under the curve

Q: Make a substitution to express the integrand as arational function and then evaluate the integral.…

A: Consider the given integration I=∫133xx2+xdx

Q: Find the integral. Show your work!

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Q: Evaluate the integral (by interpreting it as the area between two curves)

A: There are two things to be considered, if the function is odd then we will get an even function…

Q: Find the area bounded by the curves y= 5x and y = 15x.

A:

Q: find the area between the curves y=sqrt(x) , y = 1x/2 , x = 25

A: Given:

Q: Use a table of integrals to find the length of the curve y = x² from x = 0 to x = 2. %3D

A: The equation of the curve is given as, y = x2 we need to find the length of curve in the interval x…

Q: State whether it would be quicker to evaluate this integral using horizontal lines (i.e. first along…

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Q: Find the area between the curves. x = -2, x = 1, y = 2x2 + 5, y = 0

A: Given, x=-2,x=1,y=2x2+5,y=0

Q: Evaluate the integral cot Xdx

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Q: Find the area represented by the definite integral.

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Q: Find the area between the curves.

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Q: Find the area bounded by these curves.

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Q: Estimate the integral Sinx using Compostte tra pe zoidal rule with h=0.25

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Q: Find the area b0 und by Curves the y3= X-3

A: The region is bounded by y=9x2-36, y=0, x=1 and x=3

Q: Find the area between the cuo ves CUrres y=8-x² and y= -2

A: Given curve

Q: YA y= f(x)

A:

Q: Evaluate the integral using area formula:

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Q: Evaluate the definite integral that represents the area of the surface generated by revolving the cu

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Q: y = 3x2, y = 0 and x 31

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Q: Find the area bounded by the curves, 2x² + 4x + y =0 and y = 2x.

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Q: Use a double integral to find the area inside one leaf of r=3sin2*theta

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Q: Find the area bounded by the curves

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Q: * dr; By using Trapezoidal Rulc, calculate the integral

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Q: Evaluate the integral using integration by parts. cos

A: We can solve as follows

Q: Use rectangles to estimate the area under the parabola y = x2  from 0 to 1

A:

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.