Assume f is a nonnegative function with a continuous first der...

Question

Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y=f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.

Accepted Answer

Welcome back, everyone. Suppose H X is a non-negative function with a continuous first derivative on the interval 2 to 5. The curve Y equals H X is revolved around the X axis. What is the formula for the surface area of the resulting solid? First, let's make note of what we know that can help us find the formula for the surface area of this resulting solid. So far, we were told that H of X is a non-negative function. So we know that H of X is going to be greater than or equal to 0 for X in the interval from 2 to 5. Next, we're also told H of X has a continuous derivative. Continuous first derivative. On our interval from 2 to 5. And we also know that the curve Y equals H of X is revolved. Around the x-axis, OK. Now what do we know about curves that are revolved above the X-axis? Well, we know that the surface area for this curve will be equal to 2 pi. Multiplied by the integral between A and B of our function Y multiplied by the square root of 1 plus the derivative of Y with respect to X squared or with respect to X. So if we apply this to our function H of X, OK. 4 H 4 function where A equals 2, B equals 5, Y is equal to H X, and or derivative of Y with respect to X would be the derivative of H, whatever H is with respect to X. Then if we substitute that. Our surface area is going to be equal to 2 pi multiplied by the integral between 2 and 5 of H X. Multiplied by the square root of 1 plus DH. The X the derivative of H with respect to X squared or with respect to X, of course, assuming in this formula that H of X is nonnegative, so the curve lies above or on the x axis, the derivative of H X is continuous, ensuring smoothness of the surface, and that the surface areas generated by sweeping the curve are on the x axis. So this. Would be the surface area of the resulting solid. Thanks a lot for watching everyone. I hope this video helped.

Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y=f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.

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