Show that the area of the region in the first quadrant en-closed by the curve y = (1/a) cosh ax, the coordinate axes, and the line x = b is the same as the area of a rectangle of height 1 /a and length s, where s is the length of the curve from x = 0 to x = b. Draw a figure illustrating this result.

Q: Find the area of the region bounded by the parabola y = x 2  ,the tangent line to this parabola at…

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Q: Find the area of the region that is bounded by the curve ,2 - 11 sin 2 A, where r 20 and Us e 2

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Q: Find the value of k such that the area of the region under the curve y = x(k – x), (0 < x < k), is 1…

A: Given: To find the value of k such that the area of the region under the curve y=x(k-x), (0≤x≤k) is…

Q: The area enclosed by the curve y² = x(1-x) is given by .

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Q: Find the area of the region that lies inside the curve r = 1 + sin 0 and inside the curve r = 1.

A: Given: The curves  r=1+sinθ and  r=1. To Find:  Area of the region that lies inside both of the…

Q: y = cos x, y = sin 2x, 0 <x< T/2

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Q: Find the area between the curve y = cos (2x) and the line joining the points (pi/4, 0) and (pi/6,…

A: Area of the region is (2-√3)/4

Q: The exact area bounded by the curve r=sin(20) is 4 п а. 3 d. b. 2 С. П е. 2

A: The area bounded by a polar curve r=f(θ) defined over the interval θ∈α, β is given by:…

Q: Find the area of the region bounded below by the x-axis and above by the curve T æ = 2 sin?(0), y =…

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Q: Consider a region in the first quadrant bounded on the left by the y-axis, below by the line y =…

A: The given region lies in the first quadrant. It is bounded on the left by the y-axis whose equation…

Q: Find the area of the region in the first quadrant bounded by the curves y = sin x, y = cos x and the…

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Q: 10) Find the area of the region that is bounded by the given curve and lies in the specified sector.…

A: In the given question we have to find the area of the region that is bounded by the curve r=tanθ and…

Q: 8. Find the area of the region in the first quadrant bounded by the curves y = sin x, y = cos x and…

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Q: Consider the curve r = √[1+sin(θ)], where 0 ≤ θ ≤ 2π. (a) Compute the area enclosed by the graph of…

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Q: 4. Find the area of the region that lies inside the curve r = 1 – sin 0 and outside the curve r = 1.

A: It is required to find the area of the region that lies inside the curve r=1-sinθ and outside the…

Q: . Find a number k > 0 such that the area of the region bounded by the curves y = x², and y =k – x2…

A: As per as our guide line I am supposed to answer only one question from these 2 question Please…

Q: Find the area bounded by the FF. Curved and lines a.) An arch of y= (1/4)sin(x/4)

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Q: Sketch the region enclosed by the given curves and calculate its area. y= square root of x, y=0, x=4

A: Given,

Q: A// Find the area of the region bounded above by (y= e), bounded below by (y x)and the boundary line…

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Q: 5. Calculate the area contained between the curve y = cos x, -T < x < T, and the line

A: To calculate the area contained between the curve y=cos x, -π&lt;x&lt;π, and the line y=12. Note: As…

Q: Find the area of the infinite region in the first quadrant between the curve y = eX and the x-axis.

A: Given curve, y=e-xand x-axis Formula for the area of the region under a curve is given by A=∫aby dx

Q: Find the area of the region bounded by the parabola y = x², the tangent line to this parabola at (1,…

A: Given data: The equation of the parabola is y=x2. The given point is (a, b)=(1, 1). The slope of the…

Q: The area of the region bounded by the curve y= e", the x-axis,'the y-axis, and the line x 2 is equal…

A: We have to find the area of the region bounded by the curve y=e2x , the x-axis, y-axis and the line…

Q: Find the area between the curve y = sin x and the line segment joining the points (0, 0) and (5π/6,…

A: Consider: y=sin x and line joining the points 0,0 and 5π6,12. Find the area between them.   First…

Q: Find the total area of the region between the curves y = sin x and y = cos x for 0 &lt;=x&lt;=…

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Q: Find the area of the region in the first quadrant bounded by the line y = x, the line x = 2, the…

A: the area of the region in the first quadrant bounded by the line y = x, the line x = 2, the curve y…

Q: Find the area enclosed by the given curves. TU TU 26) y = sin x, y = csc 2 x, ≤x≤ 3 2 27) Find the…

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Q: Find the area of the region bounded by the parabola y=x2 , the tangent line to this parabola at…

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Q: Find the area of the region bounded by the parabola y = 2x2, the tangent line to the parabola at (3,…

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Q: Sketch the region bounded above by the curve 7e sin(Tx), the x-axis, and r = 1, and find the area of…

A: The graph of the above function is :

Q: Bm 11 2元

A: we need to find the integral of the function in the given limits to get the

Q: Find the area of the region lying between the r-axis and the curve In(3x) – 1 y = to the right of…

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Q: Find the exact area of the region inside r = 4 + 4sin(0).

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Q: The region R is bounded by the curve with equation y = sin 2x, the x-axis and the lines TT x = 0 and…

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Q: Sketch the region bounded above by the curve 8e * sin(rx), the x-axis, and I = 1, and find the area…

A: Solve for the area

Q: 1) Sketch the region enclosed by the given curves and find its area. 1 y = x for x > 0 ||

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Q: Find the area of the region lying between the æ- In(9x) – 1 axis and the curve y = to the right x2…

A: Solving by area formula

Q: Find a formula for the area A(w) of the triangle bounded by the tangent line to the graph of y = ln…

A: Given  y=ln x11y=11(lnx) We need to find out the tangent line. First we find out the slope To find…

Q: Find the area of the “triangular” region in the first quadrant bounded on the left by the y-axis and…

A: Given: Find the area of the “triangular” region in the first quadrant bounded on the left by the…

Q: Find the area of the region bounded by the parabola y = 2x², the tangent line to this parabola at…

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Q: Find the area between the graph of y = sin x and the line segment joining the points (0, 0) and ,as…

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Q: Sketch the region bounded by the curve 3xe¯", the x-axis, and x = 4, and find the area of the…

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Q: Find the area between the graph of y = sin x and the line segment joining the points (0, 0) and (4,…

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Q: Sketch the graph of the given curves and find the area of the region bounded on the left by the…

A: Graph of x=3-2y2, that is gy=3-2y2 is a parabola as function g(y) is in the form of quadratic…

Q: Find the area between the curve y = cos (2x) and the line joining the points (,0) and (등, 글).

A: Integrals and derivatives are considered as the inverses of each other. For example if we have a…

Q: Find the area of the region under the given curve from 1 to 2. 13 y = x + 4x

A: As we can calculate the area under the given by integrating the given function for the given…

Q: Calculate the area of the region bounded by the graph of f(8) = cose sine and the 0 - axis %3!…

A: The given function is f(θ)=cosθsinθ. Simplify the function. f(θ)=2sinθcosθ2=sin2θ2 The function can…

Q: Find the area between the graph of y = sin x and the line segment joining the points (0, 0) and as…

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Q: Find the area of the region enclosed by one loop of the curve. r = sin(10θ)

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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.