Posted by **Ryan** on Wednesday, November 9, 2011 at 1:45pm.

Find the area of the region bounded by the parabola y = 4x^2, the tangent line to this parabola at (4, 64), and the x-axis.

- calculous -
**Steve**, Wednesday, November 9, 2011 at 8:27pm
First, find the equation of the tangent line:

y = 4x^2

y' = 8x

slope at (4,64) = 32

(y-64)/(x-4) = 32

y = 32x - 64

32x-64 crosses the x-axis at x=2

So, we need to break the area up into two parts.

Area between the curve and y=0 on [0,2]

Area between curve and tangent line on [2,4]

Area = Int(4x^2 dx)[0,2] + Int(4x^2 - (32x-64))[2,4]

= (4/3 x^3)[0,2] + (4/3 x^3 - 16x^2 + 64x)[2,4]

= [4/3 * 8] + [4/3 * 64 - 16*16 + 64*4] - [4/3 * 8 - 16*4 + 64*2]

= 64/3

## Answer this Question

## Related Questions

calculous - Find the area of the region bounded by the parabola y = 4x^2, the ...

calculus - Find the area of the region bounded by the parabola y=x^2 , the ...

Parabola Ques - Find the point P on the parabola y^2 = 4ax such that area ...

Calc. - Find the area of the region bounded by the parabola y=x^2, the tangent ...

Math integrals - What is the area of the region bounded by y=x^2, the tangent to...

calculus help lttle question - find the area of the regin bounded by the graphs ...

Calculus - Find the volume of the solid generated by revolving the region ...

math - 1) A region is bounded by the line y = x and the parabola y = x2 - 6x + ...

math - 1) A region is bounded by the line y = x and the parabola y = x2 - 6x + ...

Math - If f(x)=3x^2-5x, find the f'(2) & use it to find an equation of tangent ...