# 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 x-axis.

1. 👍
2. 👎
3. 👁
4. ℹ️
5. 🚩
1. 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

1. 👍
2. 👎
3. ℹ️
4. 🚩

## Similar Questions

1. ### calc

1. Let R be the region bounded by the x-axis, the graph of y=sqr(x) , and the line x=4 . a. Find the area of the region R. b. Find the value of h such that the vertical line x = h divides the region R into two regions of equal

2. ### calculus

let R be the region bounded by the x-axis, the graph of y=sqrt(x+1), and the line x=3. Find the area of the region R

3. ### Calc 2

Find the area of the region bounded by the parabola y = 5x2, the tangent line to this parabola at (5, 125), and the x-axis. its not 625/3

4. ### calculus

let R be the region bounded by the graphs of y = sin(pie times x) and y = x^3 - 4. a) find the area of R b) the horizontal line y = -2 splits the region R into parts. write but do not evaluate an integral expression for the area

1. ### 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 x-axis.

2. ### Calculus

1. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6. y = x, y = 0, y = 5, x = 6 2. Use the method of cylindrical shells to find the volume V generated by

3. ### calculus

Find the volume of the solid generated by revolving the region about the given line. The region in the second quadrant bounded above by the curve y = 16 - x2, below by the x-axis, and on the right by the y-axis, about the line x =

Write the integral in one variable to find the volume of the solid obtained by rotating the firstâquadrant region bounded by y = 0.5x2 and y = x about the line x = 5. Use the mid-point rule with n = 4 to approximate the area of

1. ### Calculus: Centers of Mass

Find the centroid of the region in the first quadrant bounded by the x-axis, 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

2. ### Calculus

1. Consider the curve y = f(x) = 2^x - 1. A. Find the exact area of the region in the first quadrant bounded by the curves y = f(x) = 2^x - 1 and y = x. ("Exact area" means no calculator numbers.) B. Find the inverse function y =

3. ### Calc 2

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y = x + 2 and the parabola y = x^2 about the following lines: a) The line x=2 b) The line x=-1 c) The x axis d) The line