r = 3 sin 0 , 0 : 4. 2 (first quadrant) 2 3 (ans: 2 )

Q: 3. The centroid (#, 9) of an area D can be calculated by To dA To dA (a) Use Green's Theorem to show…

A:

Q: 29-32 Use double integration to find the area of the plane re- gion enclosed by the given curves. y…

A: Blue colour = cosh x Red colour = sinh x

Q: find the centroid of the thin plate bounded by the graphs of the given functions. Use d = 1 and M =…

A:

Q: 3 LL -3√√√9-x² Jo √9-x²-y² (x² + y² + z²) dz dy dx

A:

Q: 2. Use polar coordinate to evaluate the double integral 1 dA, + x2 + y? where D is the region in the…

A:

Q: V4-r2 Convert the integral (S" S S" rdzrdrde) to rectangular coordinates and spherical coordinates.

A: A coordinate system is a convention that is used to represent a point uniquely using a set of…

Q: Define an integral (in image) 5.1 Draw a projection on the yz plane. 5.2 Convert the integral to…

A:

Q: evaluate the double integral SRS f(r, 0) dA and sketch the region R.

A:

Q: 2. Use a double integral in polar coordinates to find the area of the region. (a) The region…

A: We need to find the area of the given regions using double integral in polar coordinates.

Q: Using polar coordinates, evaluate the integral / (2 – 3x² – 3y) dA where R is the region 4 <x + y…

A:

Q: Ex4: Find the centroid of the region that lies inside the cardioids r = a(1+ cos8) and outside the…

A:

Q: Consider the area bounded by the points P, (6,6), P2(8,4), and P3(12,4) Find the volume if it is…

A: Given area is a triangle with vertices 6,6, 8,4 and 12,4. And a line x=-3. The region with the line…

Q: Use Green’s Theorem to evaluate around the boundary curve C of the region R, where R is the triangle…

A:

Q: 4. Draw the region of integration. Then convert the following integral to polar coordinates and…

A:

Q: ) Sketch the region of Integration and write an equivalent polar integral. Then evaluate the polar…

A:

Q: Find the area enclosed by the given parametric curve and the y-axis. x = ? - 2t, y = Vt y 1.5 1.0아…

A: A=integration a to b y(t)x'(t)dt

Q: The volume of the solid that lies above the cone 0 = and below the sphere r= cos (0) is: (Here, r =…

A: Here we have to find the volume of the solid that lies above the cone θ=π6 and below the sphere…

Q: Locate the centroid y of the area. y y = a sin a ат

A:

Q: Determine the y-coordinate of the centroid of the shaded area shown. 16" Parabolic Y 5" 1 Answer: y…

A: Solution :-

Q: T/3 sin t integral S 3 csc t du dt over a region in the tu-plane, sketch the region and evaluate the…

A:

Q: a para on to express the area The portion of the plane y + 2z = 2 inside th Let u =r and v = 0 and…

A:

Q: Find the area bounded by the parametric curve x = cos(t), y = e', 0<t< 1/2, and the lines y = 1 and…

A:

Q: The region is D: x 2 0, y > 0, x² + y² < 4. Choose the sketch of the region. 4 D -4 D 4 O y 2 D 2 D…

A: The region D :x≥0,y≥0,x2+y2≤4the region is

Q: An artist has decided to finish their piece of artwork by balancing it on a fulcrum and putting it…

A:

Q: Locate the centroid of the area shown in the figure by direct integration. (Caution: Carefully…

A:

Q: Q9/ Find the centroid of the region enclosed by the cardioid r = 1 + cos 0.

A:

Q: Find the centroid of the region bounded by the graphs of the functions y = 3 sin(x), ? 1 x, and x =…

A:

Q: Using polar coordinates, evaluate the integral sin(x? + y)dA where R is the region 9 < x² + y? < 25.

A: Answer and explanation is given below...

Q: The region R shown below is bounded by the x-axis, the y-axis, and the curve given by the parametric…

A: We have to write the integral expression

Q: Find the area enclosed by the given parametric curve and the x-axis. x = 5 sin(t), y = sin(t)…

A: To find the area between the parametric curves

Q: Find the centroid of the region bounded by the graphs of the functions 1 and touching the origin. y…

A: Given curve is y=5sinx, y=17x and x=π2. We have to find the centroid x¯,y¯. The formula for the x¯…

Q: 1. Determine the centroid of the first full enclosed region in the first quadrant between sin r and…

A: given curves are                                              f(x)=sinxg(x)=cosx graph of the…

Q: Sketch the region of integration and convert the polar sum of integrals to a Cartesian integral or…

A:

Q: Consider the area bounded by the points P1 (6,6), P2(8,4), and P3(12,4) a. Find the volume if it is…

A: The given area is bounded by the points P16,6,P28,4 and P312,4. The region is shown below.  When…

Q: A. Calculate the centroid coordinates of quarter circle (x + y² a) in the first quadrant, use double…

A:

Q: 2 Determine the centroid of the area bounded by y? = 2x and the line x - y = 0. Enter your answer

A:

Q: Find the centroid of the region bounded by the graphs of the functions 1 y = 5 sin(x), y = 2, and a…

A:

Q: 89. Find the centroid of the area bounded by y = y = x³,x = 2, and the x-axis 16 =i = %3D 7

A:

Q: 5. Use spherical coordinates to evaluate the integral , z dV where E is the region between the…

A:

Q: 7 sin(æ), æ = T, I = 27 and The region R in the first quadrant of the xy-plane is bounded by the…

A:

Q: Determine the distance H from the upper surface of the symmetric double-T beam cross section to the…

A:

Q: The given curve is rotated about the y-axis. Find the area of the resulting surface. y=x²-In(x),…

A:

Q: Question 3 Let R be the region whose boundary C has parametrization r (t) = (cos t) i + (sin t) j, t…

A:

Q: Set up, but do not evaluate, an integral that represents the length of the parametric curve Select…

A:

Q: Compute the area A under the parametrized curve c(t) = (e',5t) for 0 <t < 1. %3D (Give an exact…

A:

Q: 87. Use spherical coordinates to write the differential surface areas dS, and dS, and then integrate…

A:

Q: Q.2) Find the centroid of the plane area shown below using integration. y y²= 3x y = x²/3 X

A: Given: y2=3x and y=x23 To find the centroid of the plane area  The formula for the centroid of…

Q: sec(0) Compute the area A bounded by the line r = 7 e and the rays 0 = 0, and 0 = 4 as an integral…

A: Here, the formula used is: A=12∫abr2dθ

Q: find the centroid,using polar coordinates the first quadrant area bounded by the curve r=a sin 2 θ

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.