In this video I go over Question 5 of the Discovery Project: Rotating on a Slant video series. In this question, I derive the surface area formula of the shape generated by rotating the curve about a slanted line. The derivation is fairly easy because the majority of the work was in Question 1 when the variables on the slanted line were rotated to write in terms of x.

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Discovery Project: Rotating on a Slant: Question 5

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.

Question 4

Find the volume of the solid obtained by rotating the region of Question 2 about the line y = x - 2.

Applications of Integrals: Discovery Project: Rotating on a Slant: Question 5 (Notes)