Question

Consider the region R bounded by the x-axis, y-axis, y = 4, and y = (x-3)².

a) Write, but do not evaluate, an integral which gives the volume of the solid whose base is R and whose cross-sections perpendicular to the y-axis are squares
b) Write, but do not evaluate, an integral which gives the volume of the solid formed by rotating R around the x-axis
c) Write, but do not evaluate, an integral which gives the volume of the solid formed by rotating R around the y-axis

Answer 1

The student is provided with three separate integrals to calculate the volumes of solids based on the region R, each involving different types of cross-sections or rotation about an axis, but the integrals are not evaluated.

Step-by-step explanation:

To answer the student's question involving calculus, we'll consider the region R bounded by the axes and the given functions. We will formulate, but not evaluate, integrals for three different scenarios involving solid volumes.

Volume with Square Cross-Sections

The integral for the volume of the solid whose base is R and whose cross-sections perpendicular to the y-axis are squares is given by:

∫04 (x-3)² × (x-3)² dy

Volume Formed by Rotation Around the X-Axis

The integral for the volume of the solid formed by rotating R around the x-axis using the disk method is:

π ∫016 (4 - y)² dy

Volume Formed by Rotation Around the Y-Axis

The integral for the volume formed by rotating R around the y-axis using the shell method is:

2π ∫34 x(√ y) dx