Applications of Integrals: Discovery Project: Rotating on a Slant: Question 3 (Notes)

in mathematics •  2 months ago  (edited)

In this video I go over question 3 on the Discovery Project: Rotating on a Slant video series. This time I derive the general formula for the volume of a shape generated by rotating a curve about a slanted line. I use the derivation I made earlier in Question 1 for the area of the region between the curve and the slanted line to solve for the volume. The volume equation I derive is very similar to that for the area but involves squaring the radius of revolution. To understand this question in detail, make sure to watch the first questions of this video series!

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    Discovery Project: Rotating on a Slant: Question 3

    We know how to find the volume of a solid of revolution by rotating a region about a horizontal or vertical line.

    We also know how to find the surface area of a surface of revolution if we rotate a curve about a horizontal or vertical line.

    But what if we rotate about a slanted line?

    Let C be the arc of the curve y = f(x) between the points P(p, f(p)) and Q(q, f(q)) and let R be the region bounded by C, by the line y = mx + b (which lies entirely below C), and by the perpendiculars to the line from P and Q.

    Question 1

    Show that the area of R is:

    [Hint: This formula can verified by subtracting areas, but it will be helpful throughout the project to derive it by first approximating the area using rectangles perpendicular to the line as shown above. Use the figure below to help express Δu in terms of Δx.]

    Question 2

    Find the area of the region shown in the figure below.

    Question 3

    Find a formula similar to the one in Question 1 but for the volume of the solid obtained by rotating R about the line y = mx + b.

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