Question

Let f be the function defined by f(x) = −ln(x) for 0 < x ≤ 1. R is the region between the graph of f and the x-axis.

g(x) = x^2.

Find the volume of the solid whose base is the region bounded by f(x), g(x) and the x-axis on the interval [0, 1], and whose cross-sections perpendicular to the y-axis are squares.
Your work must show the integral, but you may use your calculator to evaluate it. Give 3 decimal places for your answer.

Thank you in advance, I have tried this question several times without success.

Answer 1

To find the volume of the solid whose base is the region bounded by f(x), g(x), and the x-axis, and whose cross-sections perpendicular to the y-axis are squares, we can set up an integral.

Step-by-step explanation:

To find the volume of the solid whose base is the region bounded by f(x), g(x), and the x-axis, and whose cross-sections perpendicular to the y-axis are squares, we can set up an integral. Let's call the width of each square dx. The area of each square is g(x) squared, which is x^4. The height of each square is f(x). So, the volume of each square is x^4 * f(x) * dx. To find the total volume, we integrate this expression from x = 0 to x = 1:

Volume = ∫(x^4 * f(x) * dx) from 0 to 1

Using the given function f(x) = -ln(x), we can substitute it in the integral to get:

Volume = ∫(x^4 * -ln(x) * dx) from 0 to 1

Now, we can use a calculator to evaluate this integral and find the volume of the solid. Rounded to 3 decimal places, the volume is 0.112.

Answer 2

0.089

Step-by-step explanation:

f(x) = -ln(x) and g(x) = x²

Start by graphing the region. The two curves intersect at about (0.653, 0.426), with g(x) on the left and f(x) on the right. The region is the triangular area between the curves and above the x-axis.

If we were to cut the region horizontally (perpendicular to the y-axis), the resulting line is the width of the square cross section. The thickness of this square is dy. So the volume of the square is:

dV = A dy

dV = s² dy

dV = (x₂ − x₁)² dy

dV = (e⁻ʸ − √y)² dy

The total volume is the sum of all the squares from y=0 to y=0.426.

V = ∫ dV

V = ∫₀⁰'⁴²⁶ (e⁻ʸ − √y)² dy

Evaluate with a calculator:

V ≈ 0.089