AP Calculus AB Question

Question

Region R is bounded by the curves y = x^1/2, y= 1, and x = 4. A solid has base R, and cross-sections perpendicular to the x-axis are squares. What is the volume of this sold?

Answer 1

Does this sketch help?

Every square has a side length that is (on the bottom) equal to the red curve (y = √x) minus the blue line (y = 1) which then is (√x - 1). To find the area of some random square then, you would multiply the side length by itself, which then is (√x - 1)².

To get the area of the solid, integrate from x = 1 to x = 4:

₁∫⁴(√x - 1)²dx

which you can either choose to do manually or by using a calculator. The answer is 7/6 cubic units

Answer 2

Draw a diagram. Between x = 0 and x = 1, the graph of y = 1 lies above the graph of y = √x. When x > 1, the upper and lower boundaries are reversed.

Volume of solid = ∫_{(from 0 to 1)} (1 - √x)²dx + ∫_{(from 1 to 4)}(√x - 1)²dx