Area of Region Bounded by y = x^2 and y = 5x+6

In summary, the region bounded by y = x^2, y = 5x+6, and the negative x-axis has an area of 38.17 square units. The negative x-axis refers to the part of the x-axis to the left of (0,0). To find the area, we need to calculate the integral of the region between y= x^2 and y= 5x-6, from the point where it crosses the x-axis to (-1, 1). This is done by splitting the integral into two parts: from the point where it crosses the x-axis to (-1, 1), and from (-1, 1) to (0,0). After solving both integr
  • #1
sapiental
118
0
Consider the region bounded by y = x^2, y = 5x+6, and the negative x-axis

Compute the area of this region.

Im somewhat confused by what they mean by the negative x-axis?

The points of intersection between the two functions are [-1,1] and [6,36]

A = Integral -1 to 6 (5x-6-x^2)?

I'd appreciate if someone evaluated this problem for me step by step for I have spent several hours on it now without any clue.. Thank You!
 
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  • #2
It should be the parabola is on the top right of the region, the line is on the top left of the region, and the x-axis is on the bottom of the region
 
  • #3
sapiental said:
Consider the region bounded by y = x^2, y = 5x+6, and the negative x-axis

Compute the area of this region.

Im somewhat confused by what they mean by the negative x-axis?

The points of intersection between the two functions are [-1,1] and [6,36]

A = Integral -1 to 6 (5x-6-x^2)?

I'd appreciate if someone evaluated this problem for me step by step for I have spent several hours on it now without any clue.. Thank You!
Draw the graphs! The "negative x-axis" is the part of the x-axis left of (0,0), that corresponds to negative values of x. The integral you give is incorrect. It (with 5x+ 6, not 6x- 6) gives the area of the region between y= x2 and y= 5x- 6. The area you want has upper boundry the straight line y= 5x- 6, from the point where it crosses the x-axis to (-1, 1) where it meets the parabola y= x2, then that parabola to (0,0). The lower boundary is the x-axis, y= 0.

You will have to do it as to separate integrals.
 
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FAQ: Area of Region Bounded by y = x^2 and y = 5x+6

What is the formula for finding the area of the region bounded by y = x^2 and y = 5x+6?

The formula for finding the area of a region bounded by two curves is to take the integral of the top curve minus the integral of the bottom curve. In this case, it would be ∫(5x+6)dx - ∫(x^2)dx. This can then be solved to find the exact area of the region.

Can you explain the concept of integration and how it relates to finding the area of a region?

Integration is a mathematical tool used to find the area under a curve. It involves breaking down the curve into small, rectangular sections and adding them up to find the total area. In the case of finding the area of a region bounded by two curves, integration is used to find the difference between the two curves, which gives the total area of the region.

How do you determine the limits of integration when finding the area of a region bounded by two curves?

The limits of integration are the x-values where the two curves intersect. To find these values, set the two equations equal to each other and solve for x. These values will be the lower and upper limits of integration.

Is there a graphical representation of the area of the region bounded by y = x^2 and y = 5x+6?

Yes, the area of the region can be represented on a graph by shading in the area between the two curves. The x-values of the curves' intersection points will be the limits of integration, and the y-values will be the boundaries of the shaded region.

Can the formula for finding the area of a region bounded by two curves be applied to any two curves?

Yes, the formula can be applied to any two curves, as long as the curves intersect and the boundary of the region is clearly defined. The only difference may be in the difficulty of solving the integral, depending on the complexity of the curves.

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