In this video I continue further in the Discovery Project series which I started in my last video. In this video I use the derivation of the area of a region below a curve but above a slanted line, which I solved in my last video, to solve for an area with a known function and slanted line. In this video I solve for the area of the region bounded by the curve y = x + sin(x) and the slanted line y = x - 2. This is a useful example on using the formula for the area so make sure to watch it and stay tuned for more on this Discovery Project series!
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Discovery Project: Rotating on a Slant: Question 2
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.
