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

First, find the equation of the tangent line:
y = 4x^2
y' = 8x
slope at (4,64) = 32
(y64)/(x4) = 32
y = 32x  64
32x64 crosses the xaxis 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  (32x64))[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
Respond to this Question
Similar Questions

Calc.
Find the area of the region bounded by the parabola y=x^2, the tangent line to this parabola at (1,1) and the xaxis. I don't really get what this question is asking. It looks like the area of right triangle to me...try the graph, … 
calculus help lttle question
find the area of the regin bounded by the graphs of y=x^2=2x=3 and y=3. i don't need help solving the problem and but i am a little confused. ok the graph is a parabola and i drew a parobla with y= 3. now when find the area, am I … 
Math integrals
What is the area of the region bounded by y=x^2, the tangent to this parabola at (1, 1) and the xaxis? 
calculus
Find the area of the region bounded by the parabola y=x^2 , the tangent line to this parabola at (10, 100), and the xaxis. 
calculous
Find the area of the region bounded by the parabola y = 4x^2, the tangent line to this parabola at (4, 64), and the xaxis. 
Calculus
Find the volume of the solid generated by revolving the region bounded by the line y=5x+6 and the parabola y=x^2 about the following lines: a) the line x=6 b) the line x=1 c)the x axis d) the line y=36 Thanks! 
Parabola Ques
Find the point P on the parabola y^2 = 4ax such that area bounded by parabola, the Xaxis and the tangent at P is equal to that of bounded by the parabola, the Xaxis and the normal at P. 
calculus 2
Find the area of the region bounded by the parabola y = 3x^2, the tangent line to this parabola at (1, 3), and the xaxis. 
calculus 2
Find the area of the region bounded by the parabola y = 5x^2, the tangent line to this parabola at (3, 45), and the xaxis. 
Calculus: Centers of Mass
Find the centroid of the region in the first quadrant bounded by the xaxis, the parabola y^2 = 2x, and the line x + y = 4. I've graphed the function, and it looks like a triangle with one side curved (the parabola). I'm not quite …