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Integration help!

The tangents to the curve y=x^2– 9 are drawn at the points where the curve meets the x-axis. What is the area of the closed region bounded by the curve and the two tangents?

I've tried this again and again and still don't get the right answer!

Thanks for the help :smile:
Start by finding the points where the curve intersects the x axis, then use differentiation to find the equations of those tangents. That should get you started.
Reply 2
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maruchan
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Original post by GreenCub
Start by finding the points where the curve intersects the x axis, then use differentiation to find the equations of those tangents. That should get you started.

Yes, I got y=6x and -6x
Original post by maruchan
Yes, I got y=6x and -6x

Post your working and we can check what you've done.
Original post by maruchan
Yes, I got y=6x and -6x

Those aren't the equations of the lines themselves. You know ddxx2−9=2x\frac{\text{d}}{\text{d}x} x^2 - 9 = 2x so substituting each of the two x-values where the curve meets the x axis in place of x will give you the gradient of each of the tangents.

Then use y−y1=m(x−x1)y-y_1=m(x-x_1) to find the equation of each tangent.
Reply 5
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maruchan
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Original post by GreenCub
Those aren't the equations of the lines themselves. You know ddxx2−9=2x\frac{\text{d}}{\text{d}x} x^2 - 9 = 2x so substituting each of the two x-values where the curve meets the x axis in place of x will give you the gradient of each of the tangents.

Then use y−y1=m(x−x1)y-y_1=m(x-x_1) to find the equation of each tangent.


oh yes sorry so I get y=6x-18

do i need to use integration for this?
Original post by maruchan
oh yes sorry so I get y=6x-18

do i need to use integration for this?

Yes. You need to find the area between the x-axis and the curve between the axis crossings (integration), and the area of the triangle formed by the two lines and the x-axis.
Reply 7
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maruchan
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12
Original post by RogerOxon
Yes. You need to find the area between the x-axis and the curve between the axis crossings (integration), and the area of the triangle formed by the two lines and the x-axis.


got the answer thanks guys!
Original post by maruchan
oh yes sorry so I get y=6x-18

do i need to use integration for this?

Yep, that's one of the line equations. You don't even actually have to find the other one, because there's symmetry in the y-axis.

Sketch the graphs and use integration to find the area bounded by the axes and the tangent line between x=0 and 3 (remember that the integral comes out as negative if the area is under the y-axis - remember to make it positive). Then again use integration to find the area bounded by the axes and y=x2−9y=x^2-9 between x=0 and 3, then take the magnitude of this away from the previous area. You have to remember to double this afterwards because you only worked out half the actual area the question was asking for.

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