Answer:
the volume of the solid is 1024 cubic units.
Explanation:
1. Determine the range of the x-axis: From the given information, we know that the base of the solid is the region bounded by the x-axis, the y-axis, the graph of y = x^2, and the vertical line x = 4. So, the range of the x-axis is from 0 to 4.
2. Determine the side length of the square cross-sections: Since each cross-section perpendicular to the x-axis is a square, the side length of the square will be the same as the distance between the curves y = 0 and y = x^2.
3. Calculate the side length: To find the side length, we subtract the y-values of the curves at each x-coordinate within the given range.
- At x = 0, y = 0^2 = 0
- At x = 4, y = 4^2 = 16
So, the side length of the square cross-sections is 16 - 0 = 16 units.
4. Set up the integral: We will integrate the area of each square cross-section over the range of the x-axis (0 to 4).
- The area of each square cross-section is given by side length^2.
- The integral to find the volume is ∫[0,4] (16^2) dx.
5. Evaluate the integral: Integrate the expression 16^2 with respect to x over the range 0 to 4.
- The integral of a constant is the constant times the variable, so we have (16^2) * x evaluated from 0 to 4.
- Plugging in the values, we get (16^2) * 4 - (16^2) * 0 = 256 * 4 = 1024 units^3.