Question

Find the area of the region bounded by the parabola y = 4x², the tangent line to this parabola at (4, 64), and the x-axis.

Answer 1

To calculate the area bounded by a parabola, a tangent line at a point on the parabola, and the x-axis, find the equation of the tangent line, integrate the difference between the parabola and this line across the relevant interval, and then calculate the result to get the area.

Step-by-step explanation:

To find the area of the region bounded by the parabola y = 4x², the tangent line to this parabola at (4, 64), and the x-axis, we need to perform two steps. First, we find the equation of the tangent line, and second, we integrate the difference between the parabola and the tangent line from the appropriate bounds.

1. Determine the derivative of y = 4x² to get the slope of the tangent at any x, which is dy/dx = 8x. At x = 4, the slope is 8(4) = 32.
2. The tangent line at (4, 64) with a slope of 32 is given by the equation: y = 32x - 64.
3. Next, set up the integral of the difference between the functions from the x-intercept of the tangent (where it intersects the x-axis) to the point of tangency (x = 4): ∫ (4x² - (32x - 64)) dx from x = 2 to x = 4.
4. After integration and simplification, we get the area sought.