There is a line through the origin that divides the region bounded by the parabola y = 4x – 3x and the x-axis into two regions with equal area. What is the slope of that line?

Q: If a rectangle has its base on the r-axis and two vertices on the curve y = e-", show that the…

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Q: Find the area of the region bounded by the line y = 2x − 5 and the parabola y = x2 - 4x

A: The objective of the question is determine the area between given line and parabola.

Q: Find the area of the region bounded by the parabola y = 2x², the tangent line to this parabola at…

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Q: 5. What is the maximum vertical distance between the line y = x + 2 and the parabola y = x² for – 1…

A: The given equations are, y = x + 2       ....(1)  y = x2            ....(2) To find the maximum…

Q: t this problem?

A: Consider the given parabola:

Q: Find the minimum distance from the point (8, 3, 4) to the plane x - y + z = 4. (Hint: To simplify…

A: Let ax+by+cz+d=0 is a plane  Then distance from the  a point (xo,yo,zo) is D= |axo +byo +czo…

Q: Find the area of the region bounded by the parabola y = 3x2, the tangent line to the parabola at (3,…

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Q: What is the maximum vertical distance between the line y = x + 6 and the parabola y x2 for -2 < x<…

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Q: find the point on the line y=2x+10 that is closest to the orgin, (0,0) What is the minimum distance?

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Q: Find the slope and y-intercept of the line through the point (1,5) that cuts off the least area from…

A: In the question we have to find slope of line and y-intercept.

Q: Find the area between the curves. 3x х%3 -2, х%34, у%33е3X, у%3D2е 3X +1

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Q: What is the shortest distance from the surface ry + 15x + z2 = 204 to the origin? distance =…

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Q: Calculate the area of the smaller region (to the left of the parabola) bounded by (x + 5)2 = 2y, the…

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Q: Find the number b such that the line y = b divides the region bounded by the curves y = 16x2 and y =…

A: Given,

Q: Find the point on the line y = x closest to the point (1, 0). Hint: It is equivalent and easier to…

A: Distance between two points in xy-plane can be calculated using the distance formula. For two points…

Q: The area bounded by the parabola y = 4x2, y-axis and the lines y = 1, y = 4 is

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Q: What is the area of ​​the region between the parabola y=x^2-1 and the line y+2x=2?

A: The solution is given below in the next step:

Q: There is a line through the origin that divides the region bounded by the parabola y = 3x – 4x² and…

A: we have been given the parabola y=3x-4x2 to find the points put 3x-4x2=0⇒x(3-4x)=0⇒x=0, 34…

Q: Find the minimum distance from the point (6, 0, 9) to the plane x - y + z = 5. (Hint: To simplify…

A: Given plane is x-y+z=5 and the point is (6,0,9) normal to the plane is (1,-1,1)

Q: What is the area of the region bounded by the parabola y^2 = 2x and the line x-y-4 = 0? *

A: See the details solution in below

Q: Find the area of the region bounded by the parabola y = 4x2, the tangent line to this parabola at…

A: The equation of the parabola is given by y = 4x2.Differentiate y = 4x2 with respect to x as follows.

Q: The base of a three-dimensional figure is bound by the line x = y² and the y-axis on the interval…

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Q: Sketch the region in the xy  -plane defined by the equation or inequalities. -1 &lt;=y&lt;=3

A: -1≤y≤3

Q: Find the area of the region bounded by the parabola y = 2x2, the tangent line to the parabola at (3,…

A:

Q: Find the minimum distance from the point (6, 0, 5) to the plane x - y + z = 3. (Hint: To simplify…

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Q: y = -a(x – h)? +k (-2.1 , 2.9) (1,n)

A: The given curve is y=-ax-h2+k. Here h=-1 and k=11 given. So the curves becomes y=-ax+12+11. Also the…

Q: The area bounded by the curve y = 6x and the lines x = 7,x = 11 and x – axis ахis is

A: → y = mx is the equation of straight line passing through origin with slope m.→ Area of triangle =…

Q: What is the maximum vertical distance between the line y = x + 2 and the parabola y = x2 for -1 <x<…

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Q: There is a line through the origin that divides the region bounded by y = x − x^2 and the x-axis…

A: Area between curves

Q: Find the number b such that the line y = b divides the region bounded by the graphs of y = 9x2 and y…

A: To find the number y=b that divides the region bounded by the graphs of y=9x² and y=16 into two…

Q: Find the total area of the regions enclosed by the two curves y = 4-x2 and y = -x+2 %3D and the two…

A: The given curves are: y=4-x2, y=-x+2 and the lines x=-2, x=3

Q: What is the maximum vertical distance between the y=x+2 and the parabola y=x^2 for -1 is less than…

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Q: The area of the region enclosed by the parabola y =2 -X and the line y = - xis equal to:

A: Given that that a parabolic equation and a line equation y = -x. And we find area enclosed by…

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: Calculate the area of the smaller region (to the left of the parabola) bounded by (x+5)^2=2y , the…

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Q: Find the equation of a horizontal line that equally cuts the area bounded by the curves 2y=x^2 and…

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Q: There is a line through the origin that divides the region bounded by the parabola y = 2x – 8x and…

A: There is a line through the origin that divides the region bounded by the parabola y=2x-8x2 and the…

Q: Find the area of the region bounded by the parabola y = 4x2, the tangent line to this parabola at…

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Q: What is the average distance between the parabola y = 7x(16-x) and the x-axis on the interval…

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Q: Find the slope of the line through the point (12,15) that cuts off the least area from the first…

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Q: Find the slope and y-intercept of the line through the point (2,7) that cuts off the least area from…

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Q: Find the area of the region bounded by the parabola y = 5x2, the tangent line to this parabola a

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Q: Find the areas of the regions enclosed by the lines and curves y2 - 4x = 4 and 4x - y = 16

A: Consider a curve: y2-4x=4……1   and line: 4x-y=16……2

Q: Calculate the area of the smaller region (to the left of the parabola) bounded by (x+5)^2=2y, the…

A: Solution of the problem as follows

Q: Find the area of the region bounded by the line y = 2x – 5 and the parabola y = x² – 4x. - -

A: Graph of enclosed region -

Q: The region shown below is bounded by the curve y = 2x3 and the lines x = 8 and y = 3.

A: Given y=2x13 and lines x=8 and y=3. ax=y38when x=8⇒y=4 when y=3 ⇒x=278 By washer method,…

Q: Sketch the region in the xy  -plane defined by the equation or inequalities. |x|&lt;4 and |y|&lt;2

A: x&lt;4  and  y&lt;2⇒-4&lt;x&lt;4  and  -2&lt;y&lt;2   ∵x&lt;a⇒-a&lt;x&lt;a

Q: The area of the region enclosed by the parabola y =2-x and the line y = – xis equal to:

A:

Q: Find the parabola y = ax2 + bx + c that passes through (2,1) and be tangent to the line y = 2x + 4…

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Q: What is the maximum vertical distance between the line y = x + 6 and the parabola y = x< for -2 <x<…

A: To Find

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.