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

Answer:

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