Calculus 2 EIA 4 Due Sunday At 10:00 Pm Volumes By Washer 2

Calculus 2 Eia 4 Due Sunday At 1000pmvolumes By Washer 2 Pointsa

Using a graphing calculator, graph both functions. On the TI-84, enter the functions into Y= and then push Zoom 6. You may zoom in more by pushing Zoom Box and picking the small area in quadrant 1 and zooming in the whole screen. The intersection points are at x=0 and x=0.6, so given, there is no need to do algebra. The functions are: a) Rotate the area by the x-axis where the area is located. This gives a washer. (Part b is on page 2 below). b) Rotate the area by the x-axis with rotation center at x=0.

Paper For Above instruction

The problem involves finding the volume of a solid of revolution generated by rotating a region bounded by two functions around the x-axis. Using a graphing calculator like the TI-84, students are instructed to graph the two functions to visualize the region of interest. The intersections at x=0 and x=0.6 mark the bounds of the region. The goal is to find the volume of the solid created by rotating this area around the x-axis, which can be approached by the Washer Method.

First, students need to analyze the graph of the two functions to identify the region between x=0 and x=0.6. The specific functions are not provided in the instructions, but the student is expected to input the given functions into the calculator and identify their intersection points visually, confirming that algebraic solving is unnecessary due to the given intersection points.

Next, the critical step involves setting up the integral for the volume. When rotating the region about the x-axis, the volume is calculated using the Washer Method. The formula involves integrating the difference of the outer radius squared and the inner radius squared, multiplied by π, over the bounds of integration.

The general volume formula when rotating around the x-axis is:

V = π ∫ [f(x)^2 - g(x)^2] dx, from x=a to x=b

where f(x) and g(x) represent the outer and inner radii functions, respectively. In this case, since the rotation occurs around the x-axis and involves the regions between the functions, the student must identify which function is on top and which is below within the interval [0, 0.6].

Calculating the volume involves:

• Determining the expressions for the radii (distances from the x-axis to the functions).
• Setting up the definite integral from the bounds x=0 to x=0.6.
• Applying the Washer method formula with the correct functions in place.
• Evaluating the integral, either analytically or using calculator features.

Furthermore, ensuring all steps are shown is essential for full credit. This includes identifying the top and bottom functions over the interval, writing the integral expression explicitly, performing the integration process step-by-step, and calculating the final volume value.

This problem exemplifies the application of the Washer Method in calculus, requiring understanding of graphing, geometric interpretation, and integral setup. Proper use of the calculator's graphing features and careful analysis of the functions play vital roles in successfully solving the problem.

References

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