9:(a): Find the area boundedby the curve y =-x, the line y = x, and the line y = 2.

Q: 1.) Use the midpoint approximation method to find the area under the curve fx) = 1 - x^2over the…

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Q: Find the area between (bounded) by the Curves y=x² +10x and y= 2x + 9°

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Q: 2. Find the cerntroid of the area bounded by the curve y= 2x + 1 x+y =7 and x= 8

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Q: 8. Find the area bounded by x² – 6x + y = 0, - x² – 2x – y = 0, and the x – axis.

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Q: 5. Determine the point intersection between curve y= x(3-x)and the line y= x. Hence, find the area…

A: To find the intersection points of the given equations y = x(3-x)  and y = x

Q: Find the area bounded by the curve x = y² + 2y and the line x = 3.

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Q: x = y² and the line y = x − 2

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Q: Find the area bounded by the curve x^2 + 4y - 8 = 0, the x - axis, and the lines x = 3 and = 5

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Q: Find the area between the curve y=x^3 and its tangent through (1,1)

A: The slope of the tangent is given by y' at (x,y)=(1,1) y'=3x2At x=1y'=3 Equation of tangent line is…

Q: 16. Find the area bounded by the curve y=-x³--x²+3x and the line y=x. 16 4

A: 16)Given,y=x316-3x24+3x and y=x

Q: 1) Find the area bounded by curve y = xvx2 + 1 and the lines x = 0, x = 4 an %3D the x-axis.

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Q: Determine the area bounded by the curves x^3 - y = 0 and y + 3x^2 - 4 = 0 using a vertical element

A: The area is bounded by the given curves x3-y=0 and y+3x2-4=0. The main objective is to determine the…

Q: What is the area of the region enclosed by the curves y=e^x, y=x^2-1 and the lines x=-1 and x=1?

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Q: Compute the area bounded by the curve x=y²-2y and the line x-y-4-0₁

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Q: 1. Find the area in the second quadrant bounded by the curve 2x² + 4x +y = 0 using two methods

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Q: The area bounded by the curve z = y.y-axis and the lines Y3D0. y32 is s

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Q: Find the line y = mx that divides the area under the curve y = x(1 − x) over [0, 1] into two regions…

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Q: 7. Find the area bounded by the curve y = 4x - x^2 and the lines x = 0 and y = 4.

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Q: Find the area bounded by the curves y= sin 2x and y=cos 2x in 0,

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Q: Find the area under the curve y = 5xe- for x > 4. area =|| %3D

A: Explanation of the solution is given below...

Q: B- Find the area between the curve (y-√x - 2) and the lines (y-4-x),(y=0).

A: We have given three curves-y=x-2y=4-xy=0 First we will find intersection of first two curves…

Q: 2. Find the Area bounded by the curve y=4-x2 and the x-axis.

A: 2. The area bounded by the curve y=4-x2 and the x-axis Note:    Multiple questions are given, we are…

Q: 1. Find the area of the region enclosed by the line x = 2y and the curve x = y - y?

A: We will find out the required area.

Q: 1. Find the area bounded by the curves y – 2 = x² ,y = 6 and the y – axis ( in Quadrant I only), as…

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Q: Approximate the area under the curve y=f(x)=1/x  on the interval [1,4] , Using 3 rectangles

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Q: 15. Find the area of the region enclosed by the curve x = y³ – y' and the line x = 2y. (Hint: Need…

A: We are asked to find the area of the region

Q: find the area of the region bounded by the curve y^2=x+1 , both by the x and y axes and the line x=8

A: Using Reimann sums, the area of the region R is given by Sn=∑k=1n∆Ak. This is simply the sum of…

Q: 1. FIND THE AREA BOUNDED BY THE CURVES X= 2Jy and y %3D

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Q: Solve the area bounded by the curve x=y^2 and the line y=x+1, y=2, and y=-2. Graph manually.

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Q: Find the area of the finite region bounded by the curve y = 12 + x - x2 and the line: y = x + 3.

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Q: 10. Find the area bounded by the curve x = y² + 2y and the line x = 3.

A: Area Bounded by the Curves : If f(y) and g(y) are two continuous function on a,b and f(y)≥g(y), then…

Q: Compute for the area of the figure bounded by the curve y = x³, y = 8 and x = 0.

A: Given : y = x3 , y = 8 and x = 0. To find : Area of figure bounded by the given curve and lines?

Q: 1. Find the area under the curve y = 9- a2 over the interval[0, 3] using right end point…

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Q: 4. Find the area bounded by the curve r = 3y – y? and the line y + x 3.

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Q: 7. Find the area bounded by the curve y = x exp(-x²), the x-axis and the vertical line p through the…

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Q: Find the area bounded by the curve 2y2+2y-x-2=0 and the line x= 2y

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Q: 6. Find the area bounded by the curves of y = 6x + x² andy = -x² - 2x, and the linex = 5.

A: The equation of the curves are  C1: y = 6x+x2C2: y = -x2-2xC3: x = 5 To evaluate: The area of the…

Q: Find the area of the region bounded by the parabola y = -x² + 4x - 3 and | its tangent lines at the…

A: We have to find the area of the region.

Q: 9:(a): Find the area bounded by the curve y² = -x, the line y = x, and the line y= 2.

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Q: 1. Find the area between the curves x2-y 9 and the line y 2x-6 %3D

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Q: Find the area bounded between two curve by using Simpson' 3/8rule y=Vx+2 x=2 y = x+1

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Q: Find the area bounded by the curves y = x(x − 1)², y — axis, and the line y = 12.

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Q: Calculate the area under the curve y between x = 2 and x 8. %3D

A: Since you have posted multiple questions therefore we will answer the first one for you. If you want…

Q: Find the area bounded by the curve 4y² + 16x – 15 = 0 and the linex = -1. %3D

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Q: 1. Find the area bounded by the curves between the curve y=1+x and the line y-1 x

A: area between curve

Q: 3. Find the area of the region bounded by the curve y = Vx – 1, the y - axis and the lines y=1 and…

A: Simplify the equation y=x-1 to obtain the expression of x. y=x-1y2=x-1x=y2+1 Draw the graphs…

Q: Find the area bounded by the curves y=ex+2, y-axis, y=1, and the line x=3

A: Find the area bounded by the curves y=ex+2 and y-axis, i.e., x=0, y=1, and the line x=3. Limit for x…

Q: What is the area bounded by the curve x^2=-9y and the line y+1=0. O 3 O 4 O 5

A: Given equations: x2=-9yy+1=0

Q: Find the area bounded by the curve y = 2 over (x - 3) and the lines y = 0, x = 4 and x = 5.

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Q: Find the area between (bounded) by the curves y = x2 + 10x and y = 2x + 9.

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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.