Sketch the region enclosed by the given curves and solve for the area. y = 2 cos(4x),    y = 2 sin(8x),    x = 0,    x = π/8

Q: 1. Solve the area of the region enclosed by the graph of the equation r= 4 sin0. %3D

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Q: Find the area of the region that lies inside the first curve and outside the second curve. r = 3 +…

A: Given curves are-           C1 : π = 3 + cos(θ)            C2 : π = 4 - cos(θ)The segmented region…

Q: Find the area of the shaded region in the following graph: y 2 y = sec? y = sin(4x) T/8 T/4

A: The figure is given by To evaluate : The area of the shaded region.

Q: Sketch the curve and find the area that the curve encloses. r = 3 cos 50

A: The solution for the above question is as shown below.

Q: Find the area of the region enclosed by the inner loop of the curve. r = 2 + 4 sin ?

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Q: Evaluate the definite integral (2 + x) dx by first sketching the region and then using a geometric…

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Q: What is the area of the shaded region? y = 2 2 x = TT y = 1 + cos x

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Q: Find the area of the region enclosed between y = 2 sin(x) and y = 2 cos(x) from x = 0 to 0.47. X =…

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Q: Find the area of the region that lies inside both curves. r = 6 + 5 sin(0), r= 6 + 5 cos(0)

A: The area of the region that is to be found is graphed below. The blue curve indicates the equation…

Q: Find the area under one arch of the cycloid. x = 6a(t- sin t), y = 6a(1 – cos t)

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Q: Evaluate the integral and interpret it as the area of a region. T/2 14 sin(x) – 4 cos(2x)| dx

A: We have to

Q: Find the area shared between the curves r = cos(θ) and r^2 = 3 cos(2θ)/2

A: First of all, we will plot the graph of the given curve and identify the shared region. The given…

Q: (ii) Find the area of the region bounded by r - 200 cos (20).

A: To find- Find the area the region bounded by r2 = 200 cos2θ.

Q: Find the area of the region bounded by: r2 = 50 cos 20

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Q: 27. r = 3 cos 0, r= 1 + cos0

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Q: Find the area of the region. Interior of r = 6 sin 0

A: A detailed solution is given below.

Q: r = 3 sin 0, r=3 cos 0

A: To obtain the limit values of θ, find the point of intersection of the curves r=3sinθ and r=3cosθ.…

Q: Find the area of the region. Interior of r = 6 sin e

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Q: Find the area of the region that lies inside both curves. r= 6 + 4 sin 0, r=6 + 4 cos 0

A: The solution is given below

Q: Find the area of the region that lies inside the first curve and outside the second curve. r = 11…

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Q: Find the area of the region that lies inside the first curve and outside the second curv r = 15…

A: To find the area of the region

Q: Find the area of the region enclosed between y = 3 sin(x) and y = 3 cos(æ) from æ = 0 to a = 0.87.…

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Q: Find the area closed by the astroid ,3 x = a cosº 0, y = a sin³

A: Here we have to find the area closed by the astroid,  x=acos3θ, y=asin3θ

Q: b) Inner loop of r = 8 – 16 sin (e)

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Q: Find the area of the region that lies inside both curves. r = 3 sin(0), r = 3 cos(0)

A: Red circle is -   r=3 sin(θ) Purple circle is -   r=3 cos(θ)

Q: Find the area of the region that lies inside the curve r = 10 cos 0 and outside the curve r = 5,…

A: Given curve r=10 cos θ, r=5

Q: Find the area of the region bounded by the graphs of the equations cos? у — x, у — sin*(x), х — and…

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Q: Find the area of the region. Interior of r = 8 sin e

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Q: Evaluate the integral and interpret it as the area of a region. •T/2 14 sin(x) – 4 cos(2x)| dx

A: Given that integral ∫0π24sin(x)-4cos(2x)dx First, we have to split integral because,…

Q: Find the area of the shaded region shown in the graph. Ay y = 11 sin x y= 11 cos x 2

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Q: Find the area of the region that lies inside both curves. r = 9 sin(0), r = 9 cos(0)

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Q: What is the area of the region in the first quadrant enclosed by the graphs of y = sin (2x) and y =…

A: To find the area of the region in the first quadrant enclosed by the graphs of y=sin(2x) and y = x…

Q: TT 8. Find the area between the curves y = cos r and y = sin x, %3D

A: Between pi/4 to pi/2 sinx is greater than cosx  So y=sinx will be above y=cosx between pi/4 to pi/2…

Q: Find the area of the shaded region in the diagram given below. y = sin(2r) 6 y = cos(x)

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Q: Find the area of the region bounded by: r 98 cos 20

A: Solving

Q: Find the area of the region bounded by: 6 cos(20) 0 < 0 < 2m

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Q: Find the area of the region. Inside r = 2a cos(0) and outside r = a

A: Given:Insider=2acosθOutsider=a

Q: Find the area between the curve y = 1/x7° and the x-axis from x = 1 to co. square units

A: Given: The area between the curve y=1x76 from x=1 to ∞ To find the area of the curve. The formula…

Q: Find the area of the region that lies inside the first curve and outside the second curve. r = 1 +…

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Q: Find the area of the shaded region shown in the graph. Ay y = 27 sin x 27 y= 27 cos x

A: Solution :-

Q: Find the exact area of the region inside r = 2 sinΘ and outside r = sinΘ + cosΘ.

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Q: Find the area of the region that lies inside the first curve and outside the second curve. r= 14 cos…

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Q: Use a familiar formula from geometry to find the area of the region described and then confirm using…

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Q: . Find the area of the region enclosed the curve x = a cos³ t, y = a sin³ t. ya a a -a 0 ·a

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Q: Find the area of the region that lies inside the first curve and outside the second curve. r= 14 cos…

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Q: Also: Sketch the region whose area is given by the definite integral, and find the area by using a…

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Q: Find the area of the region that lies inside both curves. r = (3)1/2 cos(θ),    r = sin(θ)

A: Calculate the value for intersection points and the values for theta where both the value for both…

Q: Q9.a) Find the area of the region bounded by x = -/y, and y = 4 (Sketch the region) b) Solve to Find…

A: a) The region has the following graph: Thus, the area is given by…

Q: 7. r a sin 3 e

A: Explanation: Given that,  r=asin3θ  Here we will put  r=0sin3θ=0⇒3θ=0 or π⇒θ=0 or π3

Q: Find the area of the given region.

A: Consider the function: y=sin 3x·cos 2x. Find the area enclosed by the curve and x-axis in 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.