Let R be the region bounded by the a-atis, the graph of y= Vĩ, and the line (a) Find the area of the region R. (b) Find the value of A such that the vertical line = divides the region R into two regions of equal area (e) Find the volume of the solid generaled when R is revolved about the -axis. (d) The vertical line =k divides the region R ito two regions such that when these wo regioes are revolved about the -axis, they generate solids with equal volumes. Find the value of k.

Q: Use synthetic division to show that 5 is a solution of x4 - 4x3 - 9x2 + 16x + 20 = 0. Then solve the...

A: Given: The polynomial equation, x4-4x3-9x2+16x+20=0.

Q: Determine whether the statement, 'Because I have two points to work with, I use the formula for slop...

A: The statement is true.For a line, if we have two points through which the line is passing through th...

Q: Cost, revenue, and profit are in dollars and x is the number of units. If the marginal cost for a pr...

A:  

Q: Evaluate using the Squeeze Theorem.

A: Given: limx→0 x sin 1x2  To evaluate: The limit using Squeeze theorem.  

Q: Equations of tangent linesa. Find an equation of the line tangent to the following curves at thegive...

A: Given, y=cosx1-cosx;  x=π3 Part a: In order to find the tangent line at x=π3 First, we have to find ...

Q: Derivatives of constant multiples of functions Evaluate the followingderivatives

A: Consider the expression ddx-7x118 ddx-7x118=-78ddxx11=-7811x11-1=-77x108

Q: Solve ,  y"-16y= 0

A: The given differential equation is  y" - 16y = 0

Q: True or false: The domain of every rational function is the set of all real numbers.________

A: A rational function is a function of the form Rx=pxqx where p and q are polynomial function and q is...

Q: The demand function for a product is p=144−8q where pp is the price in dollars when qq units are dem...

A:

Q: Find the equation of the line normal to the graph of f(x) = x + 5x +4 at x = -1. %3D Please give you...

A:

Q: Evaluate the integrals

A: To find the integral ∫0π4sect i+tan2t j-tsint kdt

Q: Graph the curve and visually estimate its length.Then use your calculator to find the length correct...

A: Given: y=x2+x3, 1≤x≤2 To Find: i. Graph the curve and visually estimate it's length.ii. Find the len...

Q: Find an equation of the tangent line to the graph at the given point.

A: Given, fx=4xx2+6 Now we find the slope ddx4xx2+6 Now by using the quotient rule, ddxuv=ddxuv-ddxvuv2...

Q: Find the intercepts y=5x-8

A:  

Q: I need to integrate  the following equation:

A: Given,          ∫039-x2dx

Q: Use intercepts to graph the given equation : 3x + 5y + 15 = 0

A: Given equation - 3x+5y+15 = 03x+5y = -15Dividing by -15 on both sides.3x-15+5y-15 = -15-15x-5+y-3 = ...

Q: Evaluate (x+ y)ds where C is the straight-line segment x = t, y= (4- t), z=0 from (0,4,0) to (4,0,0)...

A: Given that x=t,y=4-t,z=0 from (0,4,0) to (4,0,0) evaluate ∫cx+yds  

Q: Solve the equation 11x-1 = 125

A: Given: The equation, 11x-1=125.

Q: dy Find at the given point. dx 2x e 8y = 8 at (4,0) %3D

A: Given : 2xe8y = 8

Q: Find the vertical asymptotes, if any, of the graph of the rational function: a. f(x) = x/(x2 - 1), b...

A: a. fx=xx2-1 To find the vertical asymptotes, solve the denominator by equating it to zero. That is, ...

Q: EXAMPLE 1 Radius and interval of convergence Find the radius and interval of con- vergence for each ...

A: We have to find the radius of convergence and interval of convergence for the given power series...

Q: a. Let f(x) be a function satisfying |f(x)| < x² for-1 sxs 1. Show that f is differentiable at x = 0...

A:

Q: Find the value of h if the slope of the lines represented by 6x2+2hxy+y2=0 are in the ratio 1:2

A:  

Q: if h(x) , find h' (7) to the nearest tenth. (Do not include "h'(7) 6(7*) in your answer.) %3D %3D

A: Quotient rule of differentiation  ddxfxgx=gxdfxdx-fxdgxdxgx2 We are going to use this above formula ...

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A: First calculate the derivative for the given position vector in terms of t. rt=5cost2,5sint2,12t2r't...

Q: A quadratic function has its vertex at the point (1,6). The function passes through the point (-2,-3...

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Q: In Exercises 10-37, evaluate the indefinite integral. Remember, there areno Product, Quotient, or Ch...

A: To integrate: ∫x2-sec2xdx Evaluating the integral. ∫x2-sec2xdx=∫x2dx-∫sec2xdx=x33-tanx+C

Q: Factor : x2 - 81.

A: Since the expression is  x² - 81    

Q: Derivatives Find and simplify the derivative of the followingfunctions

A:

Q: Verify that the Extended Mean Value Theorem can be applied to the functions . f(x) = ln x and g(x) =...

A: Given that, f(x) = ln x g(x) = x^3 Open Interval: (1, 4) Differentiating with respect to x we get, f...

Q: Find the vertex, focus and directrix of the parabola: (y-7)-6(x+9)

A: Given equation of the parabola is: (y - 7)^2 = 6 (x + 9) (y - 7)^2 = 4. (3/2), (x + 9) The above equ...

Q: (i) Evaluate, [2sin3xcos2: xdx. 1+3x (i) Еxpress /(х) into partial fractions. 2x² –x – 3 Hence, eval...

A: 7i∫0π22sin3xcos2xdx

Q: The following figure show the shapes and dimensions of small dams. Assuming the water level is at th...

A: If base of the dam is at y=0 and top of the dam is at y=b Then the total force on face of dam is gi...

Q: Solve and graph the solution set on a number line: 3 l2x - 1l ≥21.

A: 32x-1≥21 On simplification, we get ⇒32x-13≥213    Dividing by 3 on both sides⇒2x-1≥7⇒2x-1≤-7  or  2x...

Q: Evaluate the integral. integral    1     /   x   (x 2 +  4) 2 dx

A: Consider the integral  ∫1xx2+42dx    

Q: find the limit or explain why it does not exist

A: Consider the given limit,                   y =limx→64 x23-16x12-8 .......  (0 0 form )  So given li...

Q: The average cost of a product changes at the rate -10 1 x? 10 and the average cost of 10 units is $7...

A: Given: The average cost of a product changes at the rate C¯'(x)=-10x2+110   and the average cost of ...

Q: Find the derivative of the inverse function s(x) if f(x) = -x+ 1 1 r + 1

A:

Q: In a survey of a piece of oceanfront property, measurements of the distance to the water, d(x), were...

A: Definition used -         Trepozoid rule -              ∫abf(x)dx≈∆x2(f(x0)+2(f(x1)+f(x2)+.............

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.