The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex on the curve y = e-x2. What dimensions give the rectangle its largest area, and what is that area?

Q: A rectangle has its base on the x-axis and its upper two vertices on the parabola y = 4 – x². What…

A:

Q: Find the exact length of the curve. y = 4 + 2x3/2, 0 < x < 1

A:

Q: dr if F= and C is a piece of the parabola y=4-x² from the point (–1,3) to the point (2,0).

A:

Q: Find the exact length of the curve. 0 < x < 6 %3D 3

A: Given:   To determine: Exact length  of the curve     Concept: Exact length of the curve y = f(x)…

Q: Find the exact length of the curve. y = 3 + 6x³/2, 0≤x≤1

A: Given :y=3+6x3/20≤x≤1

Q: a) [ i) Find the exact length of the curve y = x³ + 12x 1≤x≤2

A:

Q: Consider the parabola y = -x +4x + 9 and the line y=2x + 1. Solve for the x-coordinates at their…

A: x - coordinates of the point of intersection = ?

Q: The area of the triangle formed by the tangent and the normal to the parabola y = 4ax both drawn at…

A: Given  the parabola y² = 4ax both drawn at the same end of the latus rectum and the axis of the…

Q: Find the exact length of the curve. y = 4 + 6x3/2, 0 < x < 1

A:

Q: 19. Find the area of the largest rectangle with lower base on the x- axis and upper vertices on the…

A:

Q: A point moves on the parabola 10y = 5x² + 3 in such a way that when x = 6, the abscissa is…

A: Given: A point moves on the parabola 10y=5x2+3, when x=6 the abscissa is increasing at the rate of 2…

Q: Find the exact length of the curve. y = 5 + 2x³/2, 0< x < 1

A: Given: y=fx=5+2x32, 0≤x≤1 we know that lengthL of curve in interval a,b is given by L=∫ab1+f'x2dx…

Q: Find the length of the curve: x3 1 y = - 3 4x' 1<x< 3

A:

Q: A rectangle has its two lower corners on the x-axis and its two upper corners on the parabola y = 7…

A: here the width of the rectangle is  w=2x and the right is l=7-x2

Q: Find the exact length of the curve.y = 4 + 2x3/2, 0 ≤ x ≤ 1

A: Please see the white board for the formua to calculate the length, L of the curve y = f(x), over the…

Q: Show that the number 4p is the width of the parabola x2 = 4py (p&gt;0) at the focus by showing that…

A:

Q: Find the exact length of the curve.   y = 1+ 6x3/2 , 0 ≤ x ≤ 1

A: Find the exact length of the curve.

Q: A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola…

A:

Q: Find the area of the segment cut off from the parabola x2 = 8y by the line x- 2y + 8 = U. %3D

A: Introduction ::

Q: 1. A rectangle has its base on the x-axis and its two upper corners on the parabola y= 12 – x. What…

A: The diagram for the given condition is shown below. 'A' lies on the parabola, y=12-x2 Coordinates…

Q: Find the exact length of the curve. y2 = 16(x + 3)3,   0 ≤ x ≤ 3,  y &gt; 0

A: Given: Find the exact length of the curve.

Q: (1) Find a point P in the first quadrant on the curve f(x) = such that the rectangle with sides on…

A: In the question we have to find the perimeter of the rectangle.

Q: Find the exact length of the curve. y = 2/3x3⁄2,    0 ≤ x ≤ 1

A:

Q: The region in the first quadrant bounded above by the parabola y = x², below by the x-axis and the…

A: Aa per our guideline we are allowed to do 1 question or 3 subparts at a time. Please post other…

Q: The shortest distance between the parabolas y² = x – 1 and x' = y-l is V2, then the value of…

A:

Q: 4. A coastline is given by y2 = x+3 with the ocean being to the right of the curve. If a boat crew…

A: Solution

Q: Find the point(s) on the parabola x =y closest to the point (0,3)

A:

Q: 13. Find the point on the parabola y* = 2x that is nearest the point (1, 4). %3D

A: As per our guidelines we can solve only one question or three sub-parts. To ask more please…

Q: Rotate the graph of g(x)=e^(−6x) on the interval [0,8] about the x-axis to generate a surface with…

A: Given that g(x)=e-6x To find the surface area by rotating the graph of g(x) on the interval [0,8]:

Q: A rectangle has its base on the x-axis and its upper two vertices on the parabola y=6-x² What is the…

A:

Q: Figure 4.6 shows the curves y = Vx, x= 9, y = 0, and a rectangle with its sides parallel to the axes…

A:

Q: 8. Find the function that will generate the points on a parabola that opens down, has a base of 4…

A: A parabola has base 4 and height 6

Q: Find the exact length of the curve.y = 1 + 4x3/2, 0 ≤ x ≤ 1

A: Given curve is

Q: What is the largest possible area of a rectangle inscribed in the region bounded by the parabola y =…

A: According to the figure, we have

Q: Find the exact length of the curve. y = 3 + 8x3/2, 0<x<1

A:

Q: A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 1 –…

A: first we have to draw the sketch of y = 1-x2 and draw a rectangle beneath it From the graph we can…

Q: 22=xy represents the origin. ) Prove that the equation with vertex at

A:

Q: 2. Find the dimensions of the rectangle of largest area that has its base of the x-axis and its…

A:

Q: Find the points on the curve y-9x closest to the point (0, 4). ) and (O

A:

Q: The graph of x + z² = 1 is an ordinary right circular cylinder. Indicate axis of the cylinder lies…

A:

Q: Find the exact length of the curve. x = 8 4y2' 1sys 2

A: Given, x=y48+14y2,1≤y≤2

Q: Find the x intercept(s) and the coordinates of the vertex for the parabola y = x^2 – 2x + 1. If…

A:

Q: The area of the region to the left of the parabola x=2y^2, to the right of the y-axis, and between…

A:

Q: 1. Find the point on the parabola y = 3 – x² that is closest to the point (6, 0).

A:

Q: Find the point P on the parabola y = x2 closest to the point (57, 0). (х, у) %3D ]-«

A: Given the equation of parabolay=x2         ........(1)

Q: Find the exact length of the curve. V=5+8x3/2 0<x<1

A:

Q: A rectangle has its base on the x-axis and its upper two vertices on the parabola y = 12 - x2. What…

A: First draw the situation roughly

Q: 33) Find the points of intersection between the line x - y - 2 = 0 and the parabola y = 4 - x 2.

A:

Q: Consider the graph of (x^2) - (y^2/10) - (z^2/10) =1 1.What does the cross-section in the -plane…

A: The given equation represents a hyperboloid of two sheets.

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.