The area between the two curves in the figure below is y = cos x y = sin 2x T/6 T/2

Q: Evaluate the integral (exact value). e2X sin(8x) dx =

A: We have to find the integral.

Q: Find the area between the inner and outer ovals of the curve r2 = 16(2 – cos 0)

A: Given,

Q: Find the area bounded by the trigonometric curves y=sinx and y=cosx from; x== 3.459 square units…

A: Area bounded by the curve .

Q: Find the area of the shaded region of the graph of r=1+sin (0)

A: To find → area of the shaded region of the of the graphπ = 1 + sinθ

Q: Find the arc length of the graph of y = ln(sin x) from x = π /4 to x = π/2.

A: The formula to find arc length of a curve f(x) from x=a to x=b is, L=∫ab1+f'(x)2dx.

Q: Find the circumference of the curves x=e^-2θ (sinθ+cosθ), y=e^-2θ (sinθ-cosθ) over the interval 0 ≤…

A: To find: The circumference of the given curves. Given information: Concept used: If a functions are…

Q: Find the area between the inner and outer ovals of the curve r2 16(2 – cos 0)

A: To find the area between the inner and outer ovals of the curve

Q: Find the exact area inside the curve defined by r = 2 + 3 sin 0 and outside the circle r = 2.

A:

Q: Find the area bounded by the trigonometric curves 5ñ to - 4 3.667 square units 3.459 square units…

A: topic- area bounded by curves

Q: 1. Find the area under one arch of the curve y = cos- :cosex.

A:

Q: 3. Find the area under one arch of the curve y = sin(2r).

A:

Q: Find the area of the region that lies inside r = 2 + cos 2θ but outside the curve r = 2 + sin θ

A: Given:

Q: h. y = arcsin sin (4x?) y = e? sin x + In x (arcsin z) with dz = 4 dx i. -

A: Given functions are  y=arc sin sin(4x2) y=ezsinx+|nxarcsinz With dzdx=4

Q: Find the total areas of the shaded regions

A: Required area is ∫-π03sinx1+cosx dx

Q: Find the length of the curve y = ln(sec x ), x is an element of [0, pi/4]

A:

Q: dx Sin cosx 2.

A:

Q: Question 8 Which of the following is equal to tanx dx to obtain the correct indef inite integral?…

A:

Q: Find the area inside both curves. r = 2+ sin(20),r 2+ cos (20)

A: If f is continuous and non negative on the interval α≤θ≤β with 0<β−α≤2π. Then area of the region…

Q: Find the area of the region that lies inside the curve r = 2+ cos 20, but outside the curve r = 2+…

A:

Q: x = -sin 2y x = cos y --

A:

Q: 8. Find the area enclosed by 1 leaf of r=sin30

A: Let's find.

Q: Use Simpson's rule with n = 6 to approximate %3D cos(a) dx 0.08 Keep at least 2 decimal places…

A:

Q: 7. Find the integral of the hyperbolic function below. 6 sin(x) cosh[cos(x)] dx

A: Evaluate: ∫6 sinx coshcosx dx

Q: Use a suitable substitution to find the indefinite integral S csc(sin 0) · cos(0)d0

A:

Q: Find the area inside both curves in Figure 23. r=2+ sin 20 r=2+ cos 20 FIGURE 23

A:

Q: ソ/2 Sin dx

A:

Q: By using Beta function, the integral I tan(0) d0 is

A: To find value of integral∫0π2tanθ3dθ by using Beta function. Bm,n=ΓmΓnΓm+n…

Q: What is the area of the region between the sine curve and the x-axis from to 2 ? -1 2 1 4 3.

A: Here we have to find the area of the region bounded by the sine curve and the x-axis from π2 to 2π.…

Q: Find the indefinite integral. ∫arctan x dx

A:

Q: Find the area of the shaded region of the graph of r=1+sin(0)

A: Consider the given function r=1+sinθ

Q: The integral of the given function | sin22x cos 7x dx is equal . ....

A:

Q: Evaluate the integral (exact value). e3x sin(10x) dx = %3D

A:

Q: The area between the two curves in * the figure below is y = cos x y = sin 2x T/6 T/2

A:

Q: Find the general indefinite integral

A:

Q: Find the area inside the curver = 2 cos 0 and to the right of r = sec 0.

A:

Q: Evaluate the integral (exact value). e3x sin(6x) dx %3D

A: Integration By Parts Formula

Q: 24. Find the area between the inner and outer ovals of the curve = a(1+sin 6).

A: To find- Find the area between the inner and outer ovals of the curve r2 = a21 + sin θ

Q: Find the area of the region enclosed by the curve a = 6 cos(t) – 3 sin(2t), y = 7 sin(t) with 0 <t <…

A:

Q: Evaluate the integral (exact value). eSX sin(5x) dx =

A: The given integral is,∫0πe5xsin(5x)dx

Q: The area between y = 2.sinx and y = sin2x for 0<x <T is 1 None 4

A: To find- area

Q: Find the arc length of f(x) = In(sin(x)) for - 10 8

A:

Q: Find the area inside the larger loop and outside the smaller loop of the curve + sin(0) 2

A: To find the area inside the larger loop and outside the smaller loop of the below curve. r=12+sinθ

Q: Find the trigonometric integral ∫sin2(x/2)  dx

A: From the given data:

Q: Find the trigonometric integral ∫sec4(x/2 ) dx

A: To evaluate the trigonometric integral , ∫sec4x2dx Solution: Put x2=t in the given integral we get,…

Q: 10 cos 0 and y = 10 sin³ 0, find the total length of the curve swept out by the point If x = (x, y)…

A: We have to point length of curve swept out by the point:

Q: Find the indefinite integral ∫ [arctan(x/2)] / (4 + x2) dx

A: ∫arctanx24+x2dx Now put, arctanx2=t⇒11+x22·12=dtdx⇒11+x24·12=dtdx⇒44+x2·12=dtdx⇒24+x2=dtdx⇒14+x2…

Q: The area between y = 2.sinx and y= sin2x for 0<x< T is None

A:

Q: Evaluate the integral: T/2 COS X sin (sinxax ऊंत (बिंरोकर

A: Given∫0π/2cosx sinsinx dx

Q: What is the area within the curve r = cos 5θ?

A: For the solution follow the next step.

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.