A solid S is generated by revolving the region between the x-axis and the curve y =√ sinx (0 ≤ x ≤ π) about the x-axis.(a) For x between 0 and π, the crosssectional area of S perpendicular to the xaxis at x is A(x) = _____.(b) An integral expression for the volume of S is _____ .(c) The value of the integral in part (b) is_____ .

Q: 1. Let f(x) = and let g(x) = Note that both functions have a vertical %3D x² – 4° (x – 2)² °…

A:

Q: Given f[(x - +sin y) dA, where R is the region bounded by x= 0, x=2, y=0, and R y=x. Sketch the…

A: !st we will draw graph of given equation and identify region of integral  For type1 region  y will…

Q: 4. Consider the region R bounded by the parabolas y= as depicted in the graph below. frť and y=- 3r,…

A: Area under the graph = ∫lower limitupper limitf(x)-g(x)dx

Q: Q#1. (a) Use integral to find the area between the graph of y = x³ - x and the x- axis. (b) Find the…

A:

Q: Let R be the region in the first quadrant bounded above by the graph of y = 1 + 4x and below by the…

A:

Q: 7. We have given the following curve integral: p (y² – a²) dæ + (x² +y²) dy where the edge of the…

A: Given

Q: Set up and evaluate the integral for finding the area and moments about the x- and y-axes for the…

A:

Q: x² and below by g(x) 3 2 between x = 2 and x = 4. Write the definite integral that describes the…

A: Correction: π is missing in the solution.

Q: 3. The graph below depicts f(x) = tan-'(x) between r = 0 and r = 1. arctan(x) Calculate the volume…

A:

Q: 2. Let R be the region below enclosed by the curves z = 2 − y², y = − sin (772), and x = -1, as…

A: Both cases required to divide the region into two subregions.

Q: Pls help ASAP

A: We find which of the the given areas is represented by the given integral. Note that x2+y2=a2…

Q: a- Is there any function that its integration is the same as itself. If yes, what is this function.…

A:

Q: 32) Let R be the region bounded by the graphs of f(x) = sin(x) and g(x) = x and the lines x = -1 and…

A: (32)  Height of the shell (h) = x-sin(x) Radius of the shell = x We know,V = ∫ab2πshell radius(shell…

Q: An object is created by rotating the region in the interval [0, 7] between the curve y the line y =…

A: First we have represent the given region.

Q: 7. For some microphones, the region of sensitivity takes on the shape of a rotated cardioid Consider…

A:

Q: 1. Does the definite integral [a,b]? Why or why not? 2. How would you calculate the total area of…

A: Here we have to use the concept of integration

Q: 2. The graph of f(x) = /2x – x² is a semicircle of radius 1 centered at (1,0). Draw a picture and…

A: By using simple property of circle

Q: compue the area of the region under f(x)=ln x over the interval (1,e)

A:

Q: 1. The study of integrals in calculus is motivated by the problem of calculating (a) the slope of a…

A:

Q: 65. Find the volume of the solid of revolution that results when the region under the graph of f (x)…

A: We first graph the region using a graphing calculator  This region will rotate about the x axis

Q: 2x Approximate the area under the curve y = on the interval [,2] by using the Trapezoidal Rule with…

A:

Q: VII. Given a region R enclosed by the curves TX y = 1 + sin 4 as shown below. (2,4) (2, 2) (-2,0) Y…

A:

Q: 1. Let f(x) = 3 sin x for 0 < x< T, and consider the region R bounded by the x axis and the graph of…

A: First sketch the graph on the x -y axis by plotting the values of f(x) on the y-axis and values of x…

Q: The Trapezoidal Rule ou e id b oblw Jeot en f (x) dx = h +f(x1) + f(x2) + ..+ f(xn-1) +f(xn)) b-a…

A:

Q: 3.a) Given the double integral fS,(r-y)e-v° dxdy, where D = {(x, y) : 0 <x+y < 1 & 0<x-y< 2},…

A:

Q: 36. Integral dependent only on area Show that the value of xy? dx + (x²y + 2x) dy around any square…

A:

Q: Set up, but do not evaluate, an expression involving definite integrals for the area of the region…

A:

Q: Consider the region bounded by the graphs of y = log3 x, x = 4, and r-axis whose graph is given…

A: The disc method uses the summation of the volume of cross-sectional discs having variable radii from…

Q: 72. Find the volume of the solid of revolution that results when the region under the graph of f (x)…

A:

Q: R is the region bounded above by the function f(x) = −2x+1 and below by the function g(x) = 2x/3 − 8…

A: Area bounded by two functions f(x) and g(x) over the interval [a, b] can be determined as…

Q: The sum of integrals below give the area of a region in the xy-plane. Sketch the region, label each…

A:

Q: 3. Let R be the region below bounded by y² + x² = 9 and (y-2)2 + x = 1. (0,3) (a) Set up a (sum of)…

A:

Q: (a)  Rewrite this integral so that the inner most integral is with respect to z, the middle integral…

A: In the given Integral, the limits are 0≤x≤2, x2≤y≤1, 0≤z≤x2.

Q: b) Solve the triple integral below where R is the region bounded by coordinate planes and the planes…

A:

Q: Evaluate [x] dx, using definition of integral. 0.

A:

Q: 3. Sketch the region whose area is represented by the definite integral, and evaluate the integral…

A: Given

Q: Find the area between the graph of x2 and the x-axis over the interval (0, 3). C Find the value of c…

A:

Q: Consider the function g (x) = ±x² +1. What is the approximate volume of a solid that is formed with…

A: GIVEN: g(x) = (1/4) x2 + 1

Q: 2. Consider region R below bounded by C : y = 7-1, C2 :4x = y? +8 and C3 : y = 0. Set up (do not…

A: C1:y=7-x C2: 4x=y2+8 C3: y=0

Q: a) Use the Trapezoid Rule to estimate the following integral. Use n=5 integral of (e^(x^2)) dx from…

A:

Q: a)Find the surface area of the volume generated when the curve y = x revolves around the x-axis from…

A:

Q: Find the volume of the solid of revolution that results when the region under the graph of f(x) = In…

A: Proceed as shown below

Q: Find the value of C for which the integral C de r2 +1 3x +1 converges. Evaluate the integral for…

A:

Q: The following Riemann sum gives the Right-Endpoint approximation of the area under the curve of some…

A:

Q: (a) Evaluate the integral (cosx+ (b) Find the total area between the region and x-axis…

A:

Q: Use a definite integral to find the area under the curve between the given x-values. -33 from x1 to…

A:

Q: x=8 F(x) g(x) Set up an integral to find the area A of the region enclosed between f(x) = 0.6x2 + 2…

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.