Volume Calculation Using the Disk Method

The video tutorial explains how to use the disk method to find the volume of a solid generated by rotating a region bounded by the function y = 2.3^x, between x = -1 and x = 1, about the x-axis. It covers sketching a representative rectangle, forming disks, and using the integral formula for volume calculation. The tutorial includes evaluating the integral using u-substitution and verifying the results with a graphing calculator.

What is the function that defines the region to be rotated about the x-axis?

• y = x^3

• y = 3.2^x

• y = 2.3^x

• y = x^2

What is the purpose of sketching a representative rectangle in the disk method?

• To calculate the surface area

• To determine the function's derivative

• To approximate the volume of a slice of the solid

• To find the area of the region

In the disk method, what does the radius of the disk represent?

• The diameter of the disk

• The function value at a given x

• The height of the cylinder

• The width of the rectangle

What is the formula used to calculate the volume of a right circular cylinder?

• V = 4/3πr^3

• V = πr^2h

• V = 2πrh

• V = πr^2

What is the integral setup for finding the volume of the solid using the disk method?

• ∫ from 0 to 2 of (2.3^x) dx

• π ∫ from -1 to 1 of (2.3^x)^2 dx

• ∫ from -1 to 1 of (2.3^x) dx

• π ∫ from 0 to 2 of (2.3^x)^2 dx

What substitution is used to simplify the integration process?

• u = 2x

• u = 2.3^x

• u = x

• u = ln(x)

What is the anti-derivative of 2.3 raised to the power of u?

• 1/ln(2.3) * 2.3^u

• 2.3^u

• u^2/2

• ln(2.3) * u

What is the exact volume of the solid obtained after integration?

• 9.6200

• 10.4500

• 7.8900

• 8.5300

How is the approximate volume verified?

• Using a different integration method

• By comparing with a known volume

• By re-evaluating the integral manually

• Using a graphing calculator

What is the significance of rounding the decimal approximation to four places?

• To avoid errors in integration

• To match the calculator's display

• To simplify the expression

• To ensure accuracy in calculations