8). Part 1 of 2: In the solid the base is a circle x^2+y^2=16 and the cross-section perpendicular to the y-axis is a square. Set up a definite integral expressing the volume of the solid.
Answer choices: integral from -4 to 4 of 4(16-y^2)dy, integral from -4 to 4 of (16+y^2)dy, integral from -4 to 4 (16-y^2)dy, integral from -4 to 4 of 4(16+y^2)dy, integral from -4 to 4 of 2(16-y^2)dy
9). Part 2 of 2: Evaluate the integral in the previous problem to determine the volume, V, of the solid.
Answer choices: 256, 1024/3, 1280/3, 512/3, 256/3
To find the volume of the solid in Part 1, we can use the method of cylindrical shells. Since the cross-section perpendicular to the y-axis is a square, we can consider the solid as a collection of cylindrical shells stacked on top of each other.
First, let's consider a small vertical strip of thickness dy at a distance y from the y-axis. The height of the cylindrical shell is the difference between the upper and lower edges of the square cross-section. The lower edge of the square is the circle x^2+y^2=16, which can be rewritten as x^2=16-y^2. Thus, the lower edge of the square has x-coordinates -√(16-y^2) and √(16-y^2).
The volume of the cylindrical shell is given by the product of the height, the circumference of the circle, and the thickness dy. The circumference of the circle is 2πr, where r is the distance from the y-axis to the edge of the square.
Using the Pythagorean theorem, we can find that r = √(16-y^2). Therefore, the volume of the cylindrical shell is 2π(√(16-y^2))dy.
To find the total volume of the solid, we need to integrate this expression over the appropriate range of y-values. The solid is symmetrical about the y-axis, so we can integrate from -4 to 4.
Therefore, the definite integral expressing the volume of the solid is: ∫[from -4 to 4] 2π(√(16-y^2))dy.
Now, let's evaluate this integral to determine the volume V of the solid.
Using the provided answer choices for Part 2, we can evaluate each option by substituting the limits of integration into the integral expression and simplifying.
Let's start evaluating the options one by one.
Option 1: integral from -4 to 4 of 4(16-y^2)dy
Option 2: integral from -4 to 4 of (16+y^2)dy
Option 3: integral from -4 to 4 (16-y^2)dy
Option 4: integral from -4 to 4 of 4(16+y^2)dy
Option 5: integral from -4 to 4 of 2(16-y^2)dy
By substituting the limits of integration and simplifying, we can determine which option results in the correct volume V for the solid.