Math 234-Calculus II Summer 2020

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1. Define a function \(V\) which outputs the volume of the solid formed by revolving \(g(x)\) about the \(x\)-axis for \(1 \leq x \leq b\).
2. Graph the function \(V\). Estimate \(\lim_{b\to \infty} V(b)\) from the graph.
3. Verify your answer to 1.2 by evaluating \(V(\infty)\) (type infinity to get \(\infty\)).

1. Define a function \(A\) which outputs the surface area of the surface formed by revolving \(g(x)\) about the \(x\)-axis for \(1 \leq x \leq b\).
2. Graph the function \(A\). Estimate \(\lim_{b\to \infty} A(b)\). from the graph.
3. Verify your answer to 2.2 by evaluating \(A(\infty)\).

The surface formed by rotating \(g(x)\) about the \(x\)-axis for \(1 \leq x \leq \infty\) is called Gabriel's Horn. Think of it as a bucket (turn it upright). Suppose the units are liters. What does Problem 1 tell you about how many liters of paint Gabriel's Horn holds? What does Problem 2 tell you about the amount of paint required to paint Gabriel's Horn? What is the problem here?