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Math Help - area between curve and tangent line

  1. #1
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    area between curve and tangent line

    Figure 2 shows the curve
    C with equation  y = 3x-4\sqrt{x} + 2 and the tangent to C at the point A.

    Given that A has x-coordinate 4,

    (a) show that the tangent to C at A has the equation y = 2x -2.



    The shaded region is bounded by


    C, the tangent to C at A and the positive

    coordinate axes.

    (b) Find the area of the shaded region.
    Attached Thumbnails Attached Thumbnails area between curve and tangent line-c2.jpg  
    Last edited by Tweety; April 27th 2009 at 12:37 PM.
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  2. #2
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    You need to explain in your posts what you know how to do and what you don't. This will help us help you better.

    So for (a) you need to find the derivative of your equation and construct a tangent line formula. This is fairly standard, so I'll assume you know about how to do this unless you tell me otherwise.

    Once you find this line, you need to use integration to find the area of the shaded region. First, what is the region of integration? Put another way, what are the bounds of integration? When you have an area between two curves, call it h(x), you write it in terms of the two equations. So is the region made by C-C' or C'-C?
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  3. #3
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    Quote Originally Posted by Jameson View Post
    You need to explain in your posts what you know how to do and what you don't. This will help us help you better.

    So for (a) you need to find the derivative of your equation and construct a tangent line formula. This is fairly standard, so I'll assume you know about how to do this unless you tell me otherwise.

    Once you find this line, you need to use integration to find the area of the shaded region. First, what is the region of integration? Put another way, what are the bounds of integration? When you have an area between two curves, call it h(x), you write it in terms of the two equations. So is the region made by C-C' or C'-C?
     \frac{dy}{dx} = 3-2x^{-\frac{1}{2}}

     x = 4 , 3-1=2

     y -6 = 2(x-4)
     y = 2x-2

    I still not sure how to work out the area of the shaded region, Should it be curve minus tangent line ?
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  4. #4
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    Quote Originally Posted by Tweety View Post
     \frac{dy}{dx} = 3-2x^{-\frac{1}{2}}

     x = 4 , 3-1=2

     y -6 = 2(x-4)
     y = 2x-2

    I still not sure how to work out the area of the shaded region, Should it be curve minus tangent line ?
    The area of the shaded region is given by \int_0^1 3x - 4 \sqrt{x} + 2 \, dx + \int_1^4 (3x - 4 \sqrt{x} + 2) - (2x - 2) \, dx.

    Do you understand why? (Note that the x-intercept of the tangent line is (1, 0))
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    The area of the shaded region is given by \int_0^1 3x - 4 \sqrt{x} + 2 \, dx + \int_1^4 (3x - 4 \sqrt{x} + 2) - (2x - 2) \, dx.

    Do you understand why? (Note that the x-intercept of the tangent line is (1, 0))
    Yes I get now, thanks.
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